The United States and Russia moved toward a head-spinning reset of their relationship on Tuesday, agreeing to work together on ending the Ukraine war, financial investment and re-establishing normal relations. The meeting between senior officials from both countries was a striking display of bonhomie after three years of American efforts to isolate Moscow for its 2022 invasion.

After more than four hours of talks, Secretary of State Marco Rubio said that both sides had agreed to work on a peace settlement for Ukraine as well as to explore “the incredible opportunities that exist to partner with the Russians,” both geopolitically and economically.

“We weren’t just listening to each other, but we heard each other,” Sergey V. Lavrov, Russia’s foreign minister, said. “I have reason to believe that the American side started to better understand our positions.”

The meeting, the most extensive negotiations in more than three years between the two global powers, was the latest swerve by the Trump administration in abandoning Western efforts to punish Russia for starting Europe’s most destructive war in generations. It signaled Mr. Trump’s intention to roll back the Biden administration’s approach toward Moscow, which focused on sanctions, isolation and sending weapons to Ukraine that helped kill tens of thousands of Russian soldiers.

Speaking to reporters after the meeting, American officials did not dwell on Russia’s violation of international law in attacking Ukraine, its alleged war crimes or the three years of devastation that Russian shelling and bombardment had wrought in parts of Ukraine. Instead, they repeatedly lauded Mr. Trump for trying to stop the fighting by talking to Russia in a way that his predecessor did not.

“For three years,” Mr. Rubio said, “no one else has been able to bring something together like what we saw today, because Donald Trump is the only leader in the world that can.”

The talks showed that rather than keeping the pressure on Moscow, Mr. Trump was eager to work with Russia to end the war — an approach that would most likely fulfill many of the demands of its president, Vladimir V. Putin — and that he was prepared to cast aside the worries of American allies in Europe.

The meeting came nearly a decade after Russia interfered in the 2016 election to Mr. Trump’s benefit, but then saw Mr. Trump adopt few Russia-friendly policies during his first presidency.

Tuesday’s meeting in Saudi Arabia was the latest sign that Mr. Trump’s second term could be different. In Europe and Ukraine, apprehensions are likely to deepen that the United States and Russia could try to strike their own peace deal, sidelining Kyiv and European allies.

President Volodymyr Zelensky of Ukraine reacted angrily to Tuesday’s meeting, saying he was postponing his own planned visit to Saudi Arabia — he had planned to arrive on Wednesday — to protest that Ukraine had not been invited to Tuesday’s talks.

It was important, he said that negotiations do not happen “behind the backs of the key subjects.” Any decisions, he added, “cannot be imposed” on Ukraine.

The meeting came less than a week after Mr. Trump’s lengthy phone call with Mr. Putin and took place at a palace in Riyadh, the capital of Saudi Arabia, whose crown prince, Mohammed bin Salman, has been seeking to elevate his country’s role on the world stage.

Michael Waltz, Mr. Trump’s national security adviser, and Steve Witkoff, the Middle East envoy and a longtime friend of Mr. Trump, joined Mr. Rubio for the meeting. Mr. Witkoff called the meeting “positive, upbeat, constructive.”

The Russian delegation included Yuri Ushakov, Mr. Putin’s foreign policy adviser, and Kirill Dmitriev, the head of Russia’s sovereign wealth fund. Speaking to Russian television, Mr. Dmitriev described a jovial atmosphere — “there were a lot of jokes” and a “very tasty” lunch — while Mr. Ushakov said that both sides also discussed preparations for a meeting between Mr. Trump and Mr. Putin. He offered no details on when a summit might take place.

Russia appeared to be using Tuesday’s talks to cater to Mr. Trump’s interest in profits and natural resources, arguing that American oil companies and others stood to gain hundreds of billions of dollars by again doing business in Russia.

Mr. Dmitriev, who worked with Mr. Witkoff to broker the release last week of an American schoolteacher jailed in Russia, said he would seek to restart economic cooperation with the United States to “rebuild communication, rebuild trust, rebuild success.”

“U.S. oil majors have had very successful business in Russia,” Mr. Dmitriev said in a brief interview on Tuesday before the talks began, offering an example of how the countries could rebuild business ties. “We believe at some point they will be coming back, because why would they forgo these opportunities that Russia gave them to have access to Russian natural resources?”

Russian commentators have expressed the hope that talks with the Trump administration and a peace deal in Ukraine could pave the way for the United States to lift the severe sanctions imposed by the Biden administration against Moscow.

Leading Western oil companies, including Exxon Mobil, joined many other businesses in pulling out of Russia three years ago amid outrage over Mr. Putin’s full-scale invasion of Ukraine. Mr. Dmitriev said he would present the American delegation with an estimate showing that American companies lost $300 billion by leaving Russia.

Speaking to reporters after Tuesday’s meeting, Mr. Rubio described a three-step plan for what the United States and Russia planned to do next.

First, he said, both countries would negotiate how to remove restrictions placed on each other’s embassies in Moscow and Washington, which are operating with skeleton staffs after years of tit-for-tat expulsions.

In addition, he said, the United States would engage with Russia about “parameters of what an end” to the Ukraine war would look like.

“There’s going to be engagement and consultation with Ukraine, with our partners in Europe and others,” Mr. Rubio said. “But ultimately, the Russian side will be indispensable to this effort.”

And finally, he said, Russia and the United States would explore new partnerships, both in geopolitics and in business. He described them as “the extraordinary opportunities that exist should this conflict come to an acceptable end.”

Tuesday’s discussions were the first time after Mr. Putin’s invasion in early 2022 that broad delegations of senior American and Russian officials are known to have met in person.

In Europe and Ukraine, the news of Tuesday’s talks had been met with confusion and concern. While Mr. Rubio characterized the talks as preliminary, there was widespread criticism in Europe that Mr. Trump’s approach to Russia had not been coordinated with allies of the United States.

But for Saudi Arabia’s de facto ruler, Crown Prince Mohammed, hosting the talks has offered a major opportunity to solidify his status as a global leader with influence that extends beyond the Middle East.

The Saudis, in a Foreign Ministry statement, said they were welcoming the Russians and Americans “as part of the kingdom’s efforts to enhance security and peace in the world.”

Like other countries in the Middle East, Saudi Arabia has avoided taking sides in the Ukraine war.

It has sent humanitarian aid to Ukraine while cultivating close ties with Russia. When a Ukraine peace conference was held in Switzerland last June that excluded Russia, Saudi Arabia and the neighboring United Arab Emirates refused to sign the final joint statement.

Andrew E. Kramer and Maria Varenikova contributed reporting from Kyiv, Ukraine.

Shortly after the United States’ opening meeting with Russian officials on Tuesday, President Volodymyr Zelensky of Ukraine lashed out at the Trump administration’s negotiating tactics in his harshest terms yet for excluding Ukrainians from talks on their own country’s fate.

The meeting in Riyadh ended with an agreement to establish teams to negotiate an end to the war in Ukraine and normalize relations, and with upbeat statements and pledges for closer ties between the United States and Russia — continuing a thaw in relations that Kyiv and European allies have found unnerving.

Mr. Zelensky protested his exclusion from the discussions by canceling his own planned trip to the Saudi capital.

“Decisions on how to end the war in Ukraine cannot be made without Ukraine, nor can any conditions be imposed,” Mr. Zelensky said from Turkey, where he had traveled as part of a planned tour of the Middle East. “We were not invited to this Russian-American meeting in Saudi Arabia. It was a surprise for us, I think for many others as well.”

Ukraine, he said, learned of plans for the gathering from the media. Mr. Zelensky suggested that he had intended to meet American officials after the gathering in Riyadh, the Saudi capital, on a previously scheduled state visit to Saudi Arabia.

“I don’t know who will stay, who will leave, or who is planning to go where. To be honest, I don’t care,” he said. “I don’t want coincidences, and that’s why I will not go to Saudi Arabia.”

Ukraine has been seeking talks that would provide it protection against future aggression by Russia, with a commitment of membership in NATO or peacekeepers deployed into the war zone. Ukraine has also asked nations to consider prosecutions for Russian war crimes and reparations for a conflict that has leveled whole cities and killed and wounded tens of thousands of civilians, as well as about a million soldiers on both sides.

Those kinds of demands were nowhere near the conversation in Riyadh, where American negotiators instead focused on “the incredible opportunities” that would come with an improved relationship with Moscow, according to Secretary of State Marco Rubio.

But Mr. Zelensky insisted that the terms of any settlement negotiated without Ukraine “cannot be imposed” on Ukraine.

The pointed remarks represented a shift from Mr. Zelensky, who has tried to walk a fine line in the face of Trump administration pronouncements, avoiding direct criticism. He has offered praise in recent speeches and interviews, over the weekend telling NBC that President Trump could succeed in pressuring Russia into a settlement because the Russian president, Vladimir V. Putin, feared him.

But as the meeting in Riyadh came together, he sharpened his criticism of the negotiating process.

In an interview with the German broadcaster ARD on Monday, Mr. Zelensky said the United States was seeking a quick cease-fire by “saying things that Putin really likes.” The aim of the American negotiators, he said, was to move quickly to a presidential summit with Mr. Trump and Mr. Putin and announce a truce.

“But what they want, just a cease-fire, is not success,” Mr. Zelensky said.

The United States and European nations, he has said, should first outline the terms of postwar security in Ukraine, and he has insisted that Russia accept security guarantees to prevent violations or a resumption of the war.

Mr. Zelensky compared the Russia-U.S. talks that opened in Saudi Arabia to the negotiated end to America’s military presence in Afghanistan, which opened the door to the Taliban’s return to power after a 20-year American war. The United States negotiated directly with the Taliban, cutting out the American-backed Afghan government.

“I do not think that anybody is interested in Afghanistan 2.0,” Mr. Zelensky said in an interview broadcast before Saudi talks ended. “We remember what happened in Afghanistan when the Americans left in a hurry.” That pullout, he said, was an example of “what can happen when somebody doesn’t finish, doesn’t think and leaves in a hurry.”

Mr. Zelensky has also been rebutting accusations from Mr. Putin that he is an illegitimate leader because Ukraine has not held elections (it cannot do so while it is under martial law).

Asked whether the United States supports Russia’s demand that Ukraine hold elections before any final peace settlement, Mr. Trump said it was his administration that is pressing for Ukraine to have new elections soon, not Russia.

“Yeah, I would say that, you know, when they want a seat at the table, wouldn’t the people of Ukraine have to say, like, it’s been a long time since we’ve had an election?” he said. “That’s not a Russia thing, that’s something coming from me and coming from many other countries also. You know, Ukraine is just being wiped out.”

In fact, Russia and the United States are the only notable countries calling for elections soon in Ukraine.

Mr. Trump claimed that Mr. Zelensky’s approval rating had declined because of the destruction in Ukraine and falsely suggested that the Ukrainian president was to blame for the devastation caused by Russia.

In a sign the negotiations in Riyadh brought no immediate change in how the war is being fought, antiaircraft gunfire rattled in Kyiv, Ukraine’s capital, to repel Russian exploding drones about seven hours after the discussions wrapped up.

The United States has been the largest single supplier of military and financial aid to Ukraine since Russia’s all-out invasion in 2022, though the European Union nations collectively have provided more. Several European heads of state, also excluded from the talks, convened on Monday in Paris to gauge what military assistance or peacekeeping troops European countries could commit to secure a possible cease-fire.

Ukraine depends on the United States for satellite intelligence and air defenses, including Patriot interceptors, which are its only reliable shield against Russian ballistic missiles. Ground troops are less dependent on American weaponry, as combat has evolved during the war. Ukraine’s domestically made exploding drones now inflict a majority of casualties on Russian troops.

Over the course of the war, Ukraine has fought Russia’s far larger and better equipped army to a near standstill, though momentum is now clearly in Russia’s favor. Since November 2022, about half of 1 percent of Ukrainian land has changed hands in violent but largely static combat. Russia is now creeping forward in a bloody but slow-moving offensive in the Donbas region of eastern Ukraine.

“At this point, it is clear that neither side will win this war on the battlefield,” Mr. Zelensky said on Tuesday. “Russia wanted this, it failed. No one believed in Ukraine, yet we proved ourselves and defended our independence at an incredibly high cost in the lives of our soldiers, our people. This proves that a shift toward diplomacy must happen, but it must lead to a just peace.”

Mr. Zelensky has said he hopes to reach an agreement with the Trump administration that would exchange a share of profits from natural resources for military aid. A Trump administration proposal had demanded half of the government’s proceeds from natural resources, an official familiar with the proposal said.

Mr. Zelensky had balked at the deal, saying it did not detail any security commitments from the United States in exchange.

“This is a very important issue for us, and we are highly interested in signing an agreement” with the United States, Mr. Zelensky said in a video call with reporters in Kyiv on Monday from Abu Dhabi, in the United Arab Emirates.

Mr. Zelensky, though, said that Europe was also interested in investing in Ukraine. “I told our American partners that we also have offers from Europe,” he said. Mr. Zelensky has said any deal on resources in Ukraine should consider other backers of the Ukrainian war effort.

Edward Wong contributed reporting from Washington, and Maria Varenikova from Kyiv.

The Rapid Support Forces, the paramilitary group battling for power in Sudan’s ruinous civil war, took a step toward forming its own breakaway government on Tuesday when it hosted a lavish political event in the Kenyan capital, Nairobi.

The group’s deputy leader, Abdul Rahim Dagalo, who is under American sanctions, was greeted by hundreds of cheering people as he arrived at the elaborate event, held at a state-owned convention center in downtown Nairobi.

Mr. Dagalo did not speak at the event, and a promised charter meant to pave the way for a parallel government in R.S.F.-controlled areas was not signed. Officials said they needed another three days to negotiate the terms of the charter with Abdel Aziz al-Hilu, the leader of another Sudanese rebel faction, who sat beside Mr. Dagalo.

The meeting was a moment of striking symbolism for the R.S.F., which only last month was formally accused of genocide by the United States, and comes against the backdrop of shifting battlefields in Sudan as well as a torrent of American foreign policy changes and evolving alliances in the region.

Sudan’s army has scored a series of battlefield victories in recent months, pushing the R.S.F. out of key areas in Khartoum, the capital, and in central Sudan. The R.S.F. hopes to end that losing streak, and bolster its claim to rule, by forging a government for the considerable swath of the country it holds.

In an amphitheater bedecked with Sudanese flags, where cheering men in white turbans filled entire rows, speakers railed against the army and spoke of their desire to forge a “new Sudan.”

“We need a new constitution and to draw up a new social contract that will resolve the perennial question of how Sudan is governed,” said Mr. al-Hilu, who leads a faction of the Sudan People’s Liberation Movement-North and has fought successive Sudanese governments for decades from his base in the Nuba Mountains, in southern Sudan.

Other speakers lauded the R.S.F. as a pro-democracy movement and flashed images of Lt. Gen. Mohamed Hamdan, the group’s leader, on a giant screen to loud cheers. Reports from Sudan, though, spoke of fresh atrocities by the group.

Activists and Sudanese officials accused R.S.F. fighters of killing over 200 people, including infants, during a brutal three-day assault on two villages in White Nile state, in the south of the country. Some were shot dead as they attempted to flee across the Nile River, according to Emergency Lawyers, a group that monitors the conflict.

In a statement, Sudan’s foreign ministry put the death toll at 433.

Last week in the Darfur region of western Sudan, R.S.F. fighters stormed a famine-stricken camp in the besieged city of El Fasher in an assault that killed dozens of civilians, aid groups said. The top United Nations official in Sudan, Clementine Nkweta-Salami, said she was “shocked” by the violence.

Emergency Lawyers also accused Sudan’s army of “barbaric” assaults on civilians, including killings and forced disappearances, as allied fighters hunt for R.S.F. collaborators in Khartoum.

War broke out in April 2023 when Sudan’s army and the R.S.F., whose leaders had seized power in a coup, began fighting in Khartoum. The war has torn asunder one of Africa’s largest countries and led to suffering on a sweeping scale. Fighting has caused tens of thousands of deaths, forced over 12 million people from their homes and set off a rapidly spreading famine that is likely the world’s worst in decades.

President Trump’s foreign aid freeze has deepened the pain. Hundreds of volunteer-run soup kitchens that were feeding over 800,000 people in Khartoum have closed in recent weeks as American funding dried up.

On Monday, the United Nations appealed for $6 billion to respond to the crisis.

Whether the R.S.F. plan to create its own government can succeed is uncertain, as even speakers at Tuesday’s event acknowledged. Sudan has a long history of fragile peace deals that quickly “collapsed, then returned to war,” Mr. al-Hilu told the crowd.

Still, the R.S.F. retains staunch financial and military support from its principal foreign backer, the United Arab Emirates, which appears determined to ensure that its Sudanese proxy does not lose the war, said several foreign officials who spoke on the condition of anonymity to discuss sensitive assessments.

On Feb. 8 the army chief of the Sudanese military, Gen. Abdel Fattah al-Burhan, told political leaders in Port Sudan that he also intended to create a new government. It would be composed of “independent people” and led by a new civilian prime minister, he said.

If the R.S.F. charter does come to pass, however, it could be a turning point in the war, hardening divisions and splitting the country into rival regions, much as Libya was divided after the ouster of Col. Muammar al-Qaddafi in 2011.

A portrait of President William Ruto of Kenya hung over the R.S.F. deputy leader, Mr. Dagalo, at the convention center on Tuesday.

That the R.S.F. was able to launch its political project at a state-owned convention center in Nairobi reinforced suspicions among Sudanese officials that Kenya had effectively picked a side in the conflict.

For more than a year, many Israelis and others around the world have anguished over the fates of a mother and her two young sons who were captured by gunmen and taken to Gaza during the Hamas-led assault of Oct. 7, 2023.

On Tuesday, hopes dissipated that the Israeli woman, Shiri Bibas, and her children would be returned alive when Hamas announced that it would hand over at least some of their bodies this week. Israeli officials later warned against disseminating “rumors” regarding the hostages without extensively commenting on Hamas’s statement.

For many Israelis, the kidnapping of Ms. Bibas, her husband, Yarden, and their redheaded children — Ariel, who was 4 at the time, and Kfir, then not even 9 months old — epitomized the cruelty of the Hamas-led attack that prompted the 15-month war in Gaza. The family’s capture became a rallying cry both for those who supported a deal to end the war and negotiate the hostages’ speedy release, and for those who believed Israel should continue fighting until Hamas was destroyed.

News that Hamas would turn over the bodies, part of a series of negotiated exchanges in this phase of a cease-fire agreement, followed the release of 19 living Israeli hostages in recent weeks. If those releases lifted spirits in Israel, the report of the children’s deaths left many in the country distraught.

Israel has not confirmed the deaths of the three Bibas family members, but the Israeli military said last month it was “gravely concerned” about them.

Israel is expected to release Palestinian prisoners and detainees in exchange for the bodies.

Mr. Bibas was abducted separately to Gaza. He was seen in video footage being driven away with a bleeding head wound. Ms. Bibas’s elderly parents were also killed in the Hamas-led attack.

The abduction of the Bibas family has been seared into the Israeli national psyche. The campaign for their release has featured orange balloons to symbolize the redheaded boys as well as references to Batman, a character beloved by the toddler Ariel.

In a video from the attack, Ms. Bibas could be seen desperately clutching her two sons while a Palestinian militant stood nearby. Wrapped in a blanket, she appeared terrified.

On Tuesday, the Bibas family said in a statement that Hamas’s pledge to send home their bodies had sent them “into turmoil” but that they were still awaiting further information. “Until we receive definitive confirmation, our journey is not over,” the family said.

Roughly 1,200 people were killed in the Hamas-led 2023 attack and more than 250 abducted, according to Israel. One of the hardest-hit communities was the Bibases’ hometown, Nir Oz, roughly a quarter of whose 400 residents were either killed or taken hostage. Kfir was the youngest to be seized.

The attack prompted Israel to declare war on Hamas and invade Gaza in a military campaign that killed tens of thousands of Palestinian civilians and combatants and left much of the coastal enclave in ruins.

Mr. Bibas texted other family members throughout the attack from the fortified safe room where he had hidden with his wife and children. “I love you all,” Mr. Bibas wrote as Palestinian gunmen overran Nir Oz. He later sent a final missive: “They’re coming in.”

Hamas’s armed wing, the Qassam Brigades, put out a statement in November 2023 that Ms. Bibas and her two children had been killed in an Israeli airstrike. Fears about their fates grew when they were not among the other captive mothers and children released during a weeklong cease-fire that month.

Hamas later put out a propaganda video showing Mr. Bibas in captivity sobbing as he responded to the claim his family had been killed.

A year ago, the Israeli military released footage from a security camera that it said showed Ms. Bibas and the children in Gaza on the day of their abduction being wrapped in a sheet and forced into a car. Rear Adm. Daniel Hagari, the chief spokesman for the Israeli military, said at the time that the captives had been taken to an outpost belonging to the Mujahedeen Brigades, a small armed group, in eastern Khan Younis, Gaza, and that they were then taken somewhere else.

Earlier this month, Mr. Bibas was released as part of the truce between Israel and Hamas that began in January. The agreement stipulates that Hamas release at least 33 Israeli hostages in exchange for more than 1,500 Palestinian prisoners over the deal’s first phase, which, barring an extension, is set to expire in early March.

The Israeli government said last month that Hamas had provided a list indicating that 25 of the 33 hostages were alive and that eight had been killed.

“Sadly, my family hasn’t returned to me yet,” Mr. Bibas said in a statement after his return to Israel. “They are still there. My light is still there, and as long as they’re there, everything here is dark.”

Eylon Keshet, Mr. Bibas’s cousin, described Ariel in November 2023 as a boy who loved being the center of attention and playing with toy tractors and cars. Kfir, he said, was a “chill” baby on formula who was just beginning to eat solid food.

“We are still clinging to hope,” Jimmy Miller, another relative, said in a radio interview this week. “We are hoping for tears of joy, rather than tears of sorrow.”

Pope Francis has developed pneumonia in both lungs and will not participate in activities through the weekend because of his health issues, the Vatican announced on Tuesday.

Francis, who was admitted to a hospital in Rome on Friday, had initially been diagnosed with a polymicrobial infection, which means he has a mix of microbes, like a virus or bacteria, in his lungs or another part of his respiratory tract.

In an update on his status Tuesday evening, the Vatican said that the 88-year-old pope’s clinical condition “continued to present a complex picture” and that his treatment would be modified.

The infection “arose on a picture of bronchiectasis and asthmatic bronchitis,” which required the use of cortisone antibiotic therapy, the Vatican said. A CT scan carried out Tuesday showed the “onset of bilateral pneumonia” which required additional drug therapy, the Vatican added.

“Nevertheless, Pope Francis is in good spirits,” and was grateful for the prayers for his recovery, the Vatican said.

Earlier in the day, Matteo Bruni, a Vatican spokesman, told reporters at a news briefing at the Vatican that the pope had rested, eaten breakfast and had read newspapers “as he usually does.”

The Vatican said earlier that the pope would not be attending events through the weekend. Francis delegated a Vatican prelate to participate in a Mass for him on Sunday. Last week, Francis was unable to attend another scheduled event, for artists, at the Vatican.

This is the fourth time Francis has stayed at the hospital, the Policlinico A. Gemelli.

In 2021, he had colon surgery there. In 2023, he was admitted for a respiratory infection but went home after three days. He was hospitalized again a few months later to undergo abdominal surgery for a hernia. Last February, he underwent diagnostic tests at the Gemelli hospital after a slight flu.

As a young man, he had the upper lobe of his right lung removed, and he is prone to bouts of influenza and bronchitis during the cold winter months. His medical challenges have increased in recent years. Because of knee problems and sciatica, he often uses a wheelchair, walker or cane.

Francis’ engagements have increased since the start of the 2025 Jubilee — a year of faith, penance and forgiveness of sins that takes place every quarter-century — which opened on Christmas Eve. Millions are expected to make a pilgrimage to the Vatican this year, and Jubilee events for different groups are scheduled over the next 10 months.

For years, European leaders have fretted about reducing their dependence on a wayward United States. On Monday, at a hastily arranged meeting in Paris, the hand-wringing gave way to harried acceptance of a new world in which Europe’s most powerful ally has begun acting more like an adversary.

President Trump’s plan to negotiate a peace settlement in Ukraine with President Vladimir V. Putin of Russia, with neither the Ukrainians nor Europeans invited to take part, has forced dazed leaders in capitals like Berlin, London and Paris to confront a series of hard choices, painful trade-offs and costly new burdens.

Already on the table is the possibility that Britain, France, Germany, and other countries will deploy tens of thousands of troops to Ukraine as peacekeepers. European governments are affirming the need for major increases in their military budgets — if not to the 5 percent of gross domestic product demanded by Mr. Trump, then to levels not seen since the Cold War days of the early 1980s.

“Everybody’s hyped up at the moment, understandably,” said Lawrence Freedman, emeritus professor of war studies at King’s College London. “What is clear is that whatever happens, Europe will have to step up.”

That could put its leaders in a difficult spot. While public support for Ukraine remains strong across Europe, committing troops to potentially dangerous duty on Ukrainian soil could quickly become a domestic political liability. Estimates on the size of a peacekeeping force vary widely, but under any scenario, it would be an extremely expensive undertaking at a time of straitened budgets.

President Emmanuel Macron of France, who first floated the idea of a peacekeeping force last year — to widespread skepticism in Europe — has been weakened since his decision to call parliamentary elections last summer backfired and left him with a fragile government.

Germany may not have a new coalition government for weeks after its election on Feb. 23. On Monday, its chancellor, Olaf Scholz, dismissed talk of peacekeepers as “completely premature” and “highly inappropriate” while fighting was still raging.

Prime Minister Keir Starmer of Britain, who does not have to face voters for four years, said that Britain was open to “putting our troops on the ground if necessary.” But former military officials said that after years of budget cuts, the British military was not equipped to lead a large-scale, long-term mission in Ukraine.

“Frankly, we haven’t got the numbers, and we haven’t got the equipment,” Richard Dannatt, a former head of the British Army, told the BBC. He estimated that Britain would have to supply up to 40,000 troops to a 100,000-strong force.

For some Europeans, it is too soon to talk about a post-American era on the continent. Mr. Scholz and Poland’s prime minister, Donald Tusk, warned leaders not to sunder the trans-Atlantic alliance, whatever the current tensions.

As a practical matter, a peacekeeping force would be difficult without logistical support from the United States. American security assurances, analysts said, were crucial to making it politically acceptable in European capitals, where some leaders will have to win approval from their parliaments. Mr. Starmer spoke of an “American backstop,” saying that was “the only way to effectively deter Russia from attacking Ukraine again.”

Professor Freedman said he believed that senior Trump administration officials like Secretary of State Marco Rubio and the national security adviser, Michael Waltz, understood those realities and were not bent on pulling America’s security umbrella from Europe. But he said that Mr. Trump’s goals were harder to decipher; his drive for untrammeled power at home has been deeply alarming to Europeans.

“In the past, you assumed that this was a serious, competent country,” Professor Freedman said. “It’s unnerving to think that might not be the case. There is a sense that the guardrails just aren’t there.”

At the Munich Security Conference this past week, Vice President JD Vance delivered a blistering speech in which he urged Europeans to stop shunning far-right parties and accused them of suppressing free speech.

Those comments prompted anguish among Europeans. “We have to fear that our common value base is not that common anymore,” Christoph Heusgen, who chaired the conference, said. Mr. Heusgen, who was clearly emotional at the end of his speech, later clarified that his strong feelings were because he was leaving his job and were not a reaction to Mr. Vance’s comments.

Many Germans viewed the vice president’s comments as brazen election interference. Mr. Vance, who skipped a meeting with Mr. Scholz, did find time to meet with the co-leader of the far-right Alternative for Germany party, or AfD, Alice Weidel. Germany’s mainstream parties have refused to enter coalitions with the AfD, which German intelligence agencies classify as an extremist organization.

Mr. Trump, meantime, has threatened to hit the European Union with sweeping tariffs. That could damage the bloc’s economies, which would make it even harder to lift spending on defense. NATO’s secretary general, Mark Rutte, has called on the alliance’s members to increase their spending to “considerably more than 3 percent” of gross domestic product (the United States spends 3.4 percent).

In 2023, Germany spent 1.5 percent of its gross domestic product on defense, while France spent 2.1 percent and Britain 2.3 percent.

Beyond the political and economic provocations, European leaders are struggling to make sense of the Trump administration’s strategy for Ukraine. Mr. Hegseth’s remarks signaled a reduction in American support for Ukraine’s war goals — something that European leaders regret but privately acknowledge they share.

Yet Treasury Secretary Scott Bessent, on a visit to the Ukrainian capital, Kyiv, last week, suggested that the United States could supply a “long-term security shield” for Ukraine, provided it obtained access to the country’s valuable minerals. Mr. Trump’s announcement of negotiations between him and Mr. Putin blindsided European leaders and President Volodymyr Zelensky of Ukraine.

“A contradiction runs through the United States’ approach,” Nigel Gould-Davies, senior fellow for Russia and Eurasia at the International Institute for Strategic Studies, a research group in London, wrote in an online essay. “It has signaled that the U.S. alone will negotiate an end to the war but also that Europe alone must pay for and enforce an outcome it has not played a role in deciding.”

This assumes that Mr. Trump can strike a deal with Mr. Putin. Analysts note that the United States has already granted Russia two major concessions — ruling out Ukrainian membership in NATO and suggesting that it is unrealistic for Ukraine to reclaim all its territory — without receiving anything in return.

Some liken Mr. Trump’s approach to his nuclear diplomacy with North Korea’s leader, Kim Jong-un, during his first term. Meeting Mr. Kim in Singapore, Mr. Trump gave him a valuable concession — no more military drills between the United States and South Korea — without getting a reciprocal gesture. The negotiations petered out, and North Korea has yet to give up its nuclear arsenal.

In this case, analysts said, the odds against a quick breakthrough might spare European leaders from having to commit troops, at least for now.

“Unless the position on the ground improves greatly to Ukraine’s advantage, it’s hard to imagine Russia signing up to a deal that allows large numbers of NATO troops — including British ones — on its border,” said Malcolm Chalmers, deputy director general of the Royal United Services Institute, a research group in London.

Professor Freedman said that Mr. Trump would have to persuade Mr. Putin to agree to terms that are acceptable to Mr. Zelensky — an exceedingly long shot.

“We’re a long way from the circumstances where it makes sense,” he said of a peacekeeping force. “I can’t get past the incompatibility between what Trump can offer and what the Russians want.”

Khalil al-Hayya, Hamas’s chief negotiator, said in a speech on Tuesday that militants intended to hand over the remains of four Israeli hostages to Israel on Thursday in exchange for the release of Palestinian prisoners.

Mr. al-Hayya said that members of the Bibas family — some of the best-known hostages worldwide — would be among the four bodies handed over, without saying how many. The three remaining members of the Bibas family in the Gaza Strip are Shiri Bibas and her two children.

The Israeli prime minister’s office confirmed that the bodies of four Israelis would be returned on Thursday, but officials did not respond to requests for comment about whether members of the Bibas family would be among them. The Israeli military had said until recently that there were grave concerns for the lives of Ms. Bibas and her children, though it had not confirmed their deaths.

For many Israelis, the story of the Bibas family has become a symbol of the brutality of the Hamas-led attack of Oct. 7, 2023.

Shiri Bibas was corralled by gunmen and taken to Gaza with her two red-haired children, Ariel, 4, and a baby, Kfir, who was just shy of 9 months old. Yarden Bibas, Shiri’s husband and the children’s father, was also abducted, bleeding heavily after an assailant struck his head with a hammer, relatives said. Mr. Bibas was released from captivity earlier in February.

In a statement, the Bibas family said it had not received official confirmation about the fate of Shiri and her children.

In exchange for the four bodies, Israel will release women and minors detained by Israel in Gaza after the Oct. 7 attack, according to an Israeli and a Hamas official, who spoke on the condition of anonymity to discuss the sensitive arrangement.

Mr. al-Hayya and the Israeli prime minister’s office also said the number of living hostages scheduled to be released on Saturday will be increased to six from three.

Mr. al-Hayya said Avera Mengistu and Hisham al-Sayed, Israeli citizens who have been held in Gaza for roughly a decade, would be among the six. It was not immediately clear why Hamas had decided to increase the number of living hostages to be released. The original agreement had called for Hamas to release three hostages on Saturday in exchange for Palestinian prisoners.

But one Hamas official, Mahmoud Mardawi, said in a text message that Hamas had received assurances from mediators that Israel would allow for the entry of temporary housing units and heavy machinery into Gaza before Saturday. Israeli officials did not immediately respond to a request for comment about Mr. Mardawi’s message.

On Tuesday, Israel delivered two bulldozers to Gaza after Mr. al-Hayya’s speech, said Salama Maroof, the head of the Hamas-run government media office. At least 60,000 prefabricated housing units should be delivered to Gaza during the first phase of the deal, in addition to equipment for rubble clearance, according to a copy of the agreement’s text seen by The New York Times.

Hamas has said that the homes would provide shelter for Palestinians whose homes were destroyed and that the machinery will help make it possible to retrieve bodies from the rubble.

A hostage release would indicate that implementation of the initial, six-week cease-fire agreement between Israel and Hamas is progressing, despite concerns about its fragility.

Negotiations over the second phase of the deal, which calls for a permanent end to the fighting, a full Israeli withdrawal from Gaza and the release of more hostages and prisoners, appear less solid. The two sides were to start talks over details on the next phase two weeks ago, but Qatar, a key mediating country, said the talks had yet to begin.

Hamas has accused Israel of delaying the start of the phase two discussions, while Israel still hasn’t announced publicly when it would send officials to participate in them.

As part of the first phase of the deal, Hamas is supposed to release 33 Israeli hostages, including eight who the Israeli authorities believe are dead. As of Tuesday, Hamas had freed 19 Israelis.

Shaban al-Sayed, whose son Hisham is one of the hostages set for release on Saturday, said his family was awaiting his return with deep anxiety over his condition.

Hisham crossed into Gaza in 2015, which his father said was a consequence of mental illness. Hamas has held him ever since “as a bargaining chip,” said Mr. al-Sayed, a member of Israel’s Arab Bedouin minority.

“We don’t know in what condition he’ll return,” he said. “We’re waiting for him — and when we see him, we’ll know how much we have to celebrate.”

Aaron Boxerman contributed reporting to this article.

Israel pulled its troops from towns and villages in southern Lebanon on Tuesday, according to the United Nations, but held onto strategic positions along the border as a deadline elapsed for both Israel and Hezbollah to fully withdraw from the south.

The ongoing Israeli presence on Lebanese territory risked undermining a fragile truce with the militant group Hezbollah, and raised fears in Lebanon of another prolonged Israeli occupation.

A cease-fire in November ended the deadliest war between the two sides in decades. The truce agreement stipulated that Israel and Hezbollah would cede control of southern Lebanon to the Lebanese military by the end of January — a deadline that was then extended until Tuesday.

But Israel announced on Monday that its forces would remain temporarily in five “strategic” vantage points, just across the border in Lebanese territory, until the Lebanese military fully implemented its end of the agreement, according to an Israeli military spokesman, Lt. Col. Nadav Shoshani. He declined to say how long the Israeli troops would stay there.

Under the cease-fire, Hezbollah must also pull out from southern Lebanon and the Lebanese military is supposed to deploy there. Israeli officials have accused both Hezbollah and the Lebanese army of not upholding the agreement. The Lebanese military has said it can only fully deploy in the south once Israel’s military has withdrawn.

A U.S.-led monitoring committee overseeing the truce has praised the Lebanese military’s deployment but has not released data on the extent to which Hezbollah has withdrawn its weapons and fighters.

The latest conflict began after the Hamas-led attack of Oct. 7, 2023, when Hezbollah, an Iran-backed Lebanese armed group allied with Hamas, began firing rockets on Israeli positions. Israel and Hezbollah then exchanged cross-border fire for months, until the conflict escalated into an intense war that forced over a million people from their homes.

For months, thousands of displaced Lebanese have been unable to return to areas controlled by Israel amid repeated warnings by the Israeli military to stay away.

The United Nations, which has long maintained a peacekeeping force in south Lebanon, said on Tuesday that Israel had pulled out of population centers in the south. But the organization said the continued Israeli presence in the country violated a U.N. resolution which ended the 2006 Israel-Hezbollah war and formed the basis of the recent cease-fire.

The resolution, agreed to by Israel and Lebanon, also required Hezbollah to withdraw its fighters from the south.

The delay was “not what we hoped would happen,” the U.N. Lebanon envoy and the U.N. peacekeeping force in the region said in a joint statement.

Displaced residents flocked south on Tuesday despite the continued Israeli presence, some arriving home to find their towns and villages almost unrecognizable. At least two people were injured by Israeli gunfire, according to Lebanon’s state-run news agency.

“Our memories are destroyed, and our homes,” said Yara Awada, 24, who entered Houla, a southern border town, on Tuesday for the first time in more than a year.

Most of Ms. Awada’s neighborhood is now in ruins, with her family home toppled to the ground. Her grandfather had built the house with his bare hands more than 100 years ago, she said.

“This house meant a lot to my family. Ten children grew up in this house with their cousins and relatives. Then suddenly, within seconds, everything was gone,” Ms. Awada said. “It hurts our heart to see our village like this. But we still have hope and we are going to rebuild our village better than it was.”

Ali Qassem, the mayor of the hard-hit southern Lebanese border town of Yaroun, estimated that more than 80 percent of the town had been destroyed — including homes, churches and mosques. He said the Lebanese army, which had left during Israel’s ground invasion in October, had deployed into the town early on Tuesday.

“My house here was totally destroyed,” said the mayor, adding that he still remained displaced.

Last month, Israeli forces killed more than two dozen people as they tried to enter southern border towns, according to Lebanese officials. The Israeli military said it had fired “warning shots in order to eliminate threats.”

Hezbollah’s new leader, Naim Qassem, vehemently opposed Israel’s intention to remain in parts of southern Lebanon. But in a speech on Sunday, he stopped short of threatening to resume attacks.

“Israel must withdraw completely on Feb. 18,” Mr. Qassem said. “This is the agreement,” he added. “Everyone knows how an occupation is dealt with.”

Despite their objections, Hezbollah and the Lebanese government have no real ability to compel Israel to withdraw. Lebanon’s leaders have sought to rally their Arab neighbors and appeal to the United States in bids to pressure Israel.

But experts say that Hezbollah, weakened by 14 months of conflict with Israel, is unlikely to risk renewing the conflict in the near term.

However, if Israel does remain indefinitely inside Lebanon, this could strengthen Hezbollah, regional experts said.

“If Israel stays in those five points, that is absolutely a gift to Hezbollah,” Paul Salem, the vice president for international engagement at the Middle East Institute in Washington, said. “It enables them to say the occupation cannot be ended by diplomacy, and hence that Lebanon continues to need armed resistance.”

Lebanon’s new president, Joseph Aoun, has pledged to bring all arms under the state’s control, a challenge to Hezbollah, which has long wielded inordinate influence over the country. But it is unclear how or whether Mr. Aoun can accomplish this.

The Lebanese military warned civilians on Tuesday not to approach southern towns until the military had deployed there.

The military has accused Israel of adopting a scorched-earth policy in recent weeks, including demolishing and setting fire to homes as it has pulled out of towns and villages. The Israeli military said that “the destruction of buildings was for military purposes,” adding that it was targeting Hezbollah infrastructure.

Dayana Iwaza, Sarah Chaayto, Patrick Kingsley and Johnatan Reiss contributed reporting.