The cease-fires in Gaza and Lebanon will most likely hold for now, despite being tested to their limits over the weekend, because all sides want to avoid full-scale fighting at least for a few weeks, analysts said.

In southern Lebanon, Israeli troops remained in position past the deadline on Sunday for their withdrawal, amid Israeli claims that Hezbollah had broken its own pledge to leave the area. In Gaza, Hamas failed to release a female hostage who Israel had hoped would be freed on Saturday, prompting Israel to delay the agreed return of displaced Palestinians to their homes in northern Gaza.

But even as each side accused the other of reneging on their deals, analysts said, both Israel and its opponents had reasons to remain flexible and temporarily overlook the other’s transgressions.

Hezbollah, though angry at Israel for keeping troops in southern Lebanon, would risk a devastating Israeli counterattack if it renews its rocket strikes on Israeli cities. Hamas wants to retain power in Gaza and risks losing it if war returns. And Israel needs to maintain the current arrangement in Gaza long enough to free at least two dozen more hostages. Israeli leaders have also appeared eager to placate President Trump, who campaigned on a promise to keep peace in the Middle East.

Illustrating their desire to prolong the Gaza truce, Israel and Hamas seemed to resolve the weekend’s crisis close to midnight on Sunday. The government of Qatar, a mediator between the sides, said that the female hostage, Arbel Yehud, would be freed this week along with two others who would be released ahead of schedule. In return, Israel said that it would allow displaced Palestinians to return to northern Gaza on Monday morning. By Monday night, more than 300,000 Gazans had returned, the Hamas-run government media office said.

As for Lebanon, the White House announced that the truce there would be extended until Feb. 18, though there was no immediate comment from Israel or Hezbollah. The Lebanese prime minister’s office confirmed the extension.

Aaron David Miller, a former U.S. negotiator in Middle East peace talks, said, “They’re going to get through the next few weeks — beyond that is anyone’s guess.”

“These are not agreements between the United States and Switzerland. They’re agreements that depend on each side giving the other a certain discretion and margin for maneuver,” he added. “That is their weakness, but also their strength.”

That wriggle room ultimately allowed both truces to survive the weekend, even as Israeli troops shot and killed people in both Lebanon and Gaza who were trying to return to areas still controlled by Israel.

The Lebanese Health Ministry said that 22 people had been killed by Israeli fire in southern Lebanon, and the Palestinian Authority’s news agency said that one person had been killed in Gaza as large crowds in both places gathered near Israeli troops, demanding to go home.

But by Monday morning, the standoff in Gaza appeared set to ease. In Lebanon, Hezbollah issued a statement praising the residents attempting to return and calling on foreign powers to force Israel to withdraw. But Hezbollah did not resume its rocket fire.

Analysts say that Hezbollah is unlikely to risk further losses while its leadership is decimated and its benefactor, Iran, is weakened. Also, the group’s main arms supply route, through Syria, was blocked in December when the Syrian president, Bashar al-Assad, an ally of Hezbollah, was ousted by rebels.

Hezbollah’s commanders do “still have some rockets, they have some guns, they can do something,” said Hanin Ghaddar, a Lebanese analyst at the Washington Institute for Near East Policy, a foreign affairs research group.

“But it’s suicidal if they do this, because they know that any kind of attack by Hezbollah in Israel means that Israel will take the opportunity to come back full blast and annihilate whatever is left of them,” Ms. Ghaddar added.

Hezbollah is also probably wary of losing support among its Shiite Muslim base, particularly in next year’s parliamentary elections, Ms. Ghaddar said. Lebanon’s Shiite community paid the largest price for Hezbollah’s decision to go to war with Israel in October 2023 in solidarity with its ally Hamas. Shiite villages and towns in southern Lebanon bore the brunt of Israel’s ensuing air campaign and ground invasion.

“If the Shia do not vote for them, this is the end of Hezbollah,” said Ms. Ghaddar, the author of a book about Hezbollah’s relationship with its base. “They cannot really do anything if they don’t know 100 percent that the Shia community is going to support it.”

Because Hezbollah is less likely to resume fighting, the Gaza cease-fire is considered the frailest of the two truces.

But its biggest stress test is not expected until the beginning of March, when Hamas and Israel must decide whether to extend the arrangement beyond an initial 42-day truce.

For now, Israel has signaled it wants to maintain the cease-fire to sustain the flow of hostage releases. But an extension would require both sides to agree to a permanent end to the war — a bridge that Prime Minister Benjamin Netanyahu of Israel has appeared unwilling to cross. Mr. Netanyahu’s coalition government relies on far-right lawmakers who seek permanent Israeli control of Gaza, and his administration could collapse if the war ends with Hamas still in charge.

The terms of the agreement allow for some flexibility. The truce can continue beyond the 42-day mark as long as both sides remain negotiating about whether to make the arrangement permanent.

But Israeli officials say they will not remain locked in endlessly fruitless negotiations, especially if Hamas stops releasing hostages. And Hamas is unlikely to keep freeing the hostages, its main bargaining chip, without an Israeli promise to cease hostilities permanently.

“Hamas wants a cease-fire but not at all costs,” said Mkhaimar Abusada, a Palestinian political scientist from Gaza. “They want a cease-fire that ends the war.”

Much could depend on President Trump’s willingness to cajole Mr. Netanyahu toward a more lasting truce. Mr. Trump’s private messages to the Israeli prime minister were crucial to the forging of the initial phase, but it remains to be seen whether the American president will maintain that position beyond a few weeks.

“If Netanyahu succeeds in convincing Trump of the need to renew the war, there’ll probably be a renewal of the war,” Mr. Abusada said. “If Trump keeps his promise that he doesn’t want any wars and he wants more peace — whether it’s in Gaza, Ukraine or all over the world — that’s a different thing.”

A rebel militia backed by Rwanda on Monday announced the capture of the city of Goma in eastern Democratic Republic of Congo, a major victory for the group and one of the most significant escalations in the conflict between the two countries in years.

The militia, known as M23, briefly occupied Goma once before, in 2012, then was defeated and lay dormant for almost a decade. Now it has come roaring back, aiming to occupy the region for the long term and exploit its valuable rare minerals, with the backing of several thousand Rwandan troops who are in Congo, according to United Nations experts.

This time, M23 appears to be in a stronger position to keep hold of Goma, a city made up mainly of people who left their homes in terror and will now have to live under the rule of one of the armed groups they fled.

A spokesman for M23 announced the “liberation of the city of Goma” in a post on X, saying Congolese military personnel had to hand over their arms to the U.N. and assemble in a stadium before 3 a.m.

The United States and France both acknowledged that Goma had been captured by M23, according to two Western diplomatic and humanitarian sources, though they had yet to announce it publicly.

On Sunday, Congo’s foreign minister, Thérèse Kayikwamba Wagner, said that more Rwandan troops had crossed the border, coming directly into Goma. This amounted to “a declaration of war” on Rwanda’s part, she said.

On Monday morning, gunfire rang out over Goma. A line of military pickups lay abandoned on a road running through part of the city. Residents said that Congolese soldiers had left them there and fled on boats over Lake Kivu.

Videos circulated in local WhatsApp groups of rebels marching through the city and of a few citizens lining the streets in welcome, in what observers said was probably an attempt to placate the newly occupying militia. Reports circulated that Goma’s prison had been opened and its inmates released, but there was no independent confirmation of that move.

The conflict in eastern Congo — an area about the size of Michigan — was once labeled Africa’s World War. It has been going on since the 1990s, and has involved dozens of armed groups, of which M23 is currently dominant.

The rebels plan to occupy and exploit the region for the long term, according to the U.N. and the United States, which say the group is funded and directed by Congo’s much smaller neighbor Rwanda. Rwanda denies those accusations.

The rebels’ advance on Goma, which began with an offensive in the region launched this month, escalated rapidly over the past three weeks, prompting people to flee to — and from — the city.

On Sunday morning, thousands of people arrived in Goma from areas north of the city, some toting the few possessions they had managed to grab in pieces of cloth tied around their foreheads, others carrying babies only a few days old. Many had already been displaced and were fleeing bombs that had fallen near camps. Others had left their villages, which were caught in the crossfire between M23 and the Congolese armed forces.

Camps on the outskirts of Goma that had hosted more than 300,000 people completely emptied out in a few hours, the U.N. said.

On Friday, the military governor of North Kivu, the province of which Goma is the capital, was fatally wounded on the battlefield, according to a Congolese military spokesman. The circumstances of his death remained murky, but the spokesman said that the governor, Gen. Peter Cirimwami, died as he was being evacuated to Kinshasa, Congo’s capital.

On Saturday, the United Nations peacekeeping mission in Goma evacuated many of its staff members from the city, putting them on buses to the airport with plans to fly out to Entebbe, in neighboring Uganda.

In 2012, Rwanda came under intense international pressure to stop backing M23, and as a result, the militia was defeated the following year. But it is unclear whether such pressure can be summoned again, experts said. Rwanda has built up its relationships with Western nations since then, and has become less dependent on aid.

At an emergency meeting of the United Nations Security Council on Sunday, the United States and other members condemned Rwanda’s actions, but stopped short of calling for sanctions.

Bintou Keita, the head of the U.N. peacekeeping mission in Congo, called for Rwandan armed forces to withdraw from Congo. She told the meeting that three peacekeepers had been killed trying to protect Goma and a nearby town, Saké, from M23’s advance. She also said that the rebels had closed the airspace over Goma.

“In other words, we are trapped,” she said.

As the rebels advanced on Goma, an already dire humanitarian situation was becoming even worse. Over 400,000 people have fled their homes since the start of this year, according to the U.N. refugee agency, as M23 rebels have attacked new areas of North Kivu province, where Goma is, and South Kivu. They joined 4.6 million people who were already displaced in Congo’s east.

And still, people poured into Goma, often in long columns.

Some pushed wheelbarrows with a few salvaged belongings. Some had bicycles or carried mattresses on their heads and backs. Many of them had life-threatening injuries.

Myriam Favier, the head of the International Committee of the Red Cross’s sub-delegation in Goma, said on Friday that the day before, more than 100 people had arrived within 24 hours at the hospital where she worked — normally the number of people who arrive in a whole month.

“They’re coming from everywhere,” she said. “They’re coming from all fronts.”

Ms. Favier described medical staff treating patients with mortar or shrapnel wounds and said that the number of minors with serious injuries had been increasing significantly. She called on those using heavy artillery to reduce their attacks, saying that so many people were arriving with head wounds and chest trauma that the hospital had run out of beds and had to put patients on mattresses in the parking lot.

Across Goma, schools were being turned into shelters for displaced people. Families were stocking up what supplies they could so they did not have to venture out.

Many of the people who fled had sought sanctuary in Goma, knowing it was in the rebels’ sights, but having no other option.

As the city fell into M23’s hands, they hid where they could, many of them hungry, cold or badly injured. Some slept in the street, others in hospitals.

Solange Safi Ndakwinja was trying to look after her three daughters, who were badly injured by a bomb that exploded at an army checkpoint.

“My hope is God will help us,” Ms. Ndakwinja said. “For the rest, we don’t know what will happen.”

Elian Peltier contributed reporting from Dakar, Senegal, and Declan Walsh from Nairobi, Kenya.

They have been circling each other carefully for seven days now — sending out invitations to talk, mixing a few jabs with ego-stroking, suggesting that the only way to end the Ukraine war is for the two of them to meet, presumably without the Ukrainians.

President Trump and Vladimir V. Putin, whose relationship was always the subject of mystery and psychodrama in the first Trump term, are at it again. But it is not a simple re-run. Mr. Trump was unusually harsh in his rhetoric last week, saying Mr. Putin was “destroying Russia,” and threatening sanctions and tariffs on the country if it doesn’t come to the negotiating table — a fairly empty threat given the tiny amount of trade between the U.S. and Russia these days.

Calculating and understated as ever, Mr. Putin has responded with flattery, agreeing with Mr. Trump that Russia would not have invaded Ukraine had Mr. Trump been president three years ago. He repeated that he was ready to sit down and negotiate over the fate of Europe, superpower to superpower, leader to leader.

So far they have not spoken, though Mr. Trump told reporters on Air Force One on Saturday night that “he wants to speak, and we’ll be speaking soon.” As they prepare the ground for that first conversation, they are sending signals that they want to negotiate about more than just Ukraine — a war that, in Mr. Putin’s telling, is only one of the arenas in which the West is waging its own fight against Russia.

Both men seem to envision taking on the whole relationship between Moscow and Washington, possibly including revived nuclear arms talks, a conversation that has a looming deadline: The major treaty limiting the arsenals of both nations expires in almost exactly a year. After that, they would be free to pursue the kind of arms race the world has not seen since the deepest days of the Cold War.

Recalling conversations with Mr. Putin in 2020, before his defeat in the U.S. election that year, Mr. Trump insisted last week, “We want to see if we can denuclearize, and I think that’s very possible.” He appeared to be assuming that China would engage in the same conversation. (It has refused, at least so far.)

While he kept using the word “denuclearize,” Mr. Trump almost certainly meant negotiating a new agreement to reduce — not eliminate — the stockpiles of strategic nuclear weapons, which can cross continents. For his part, Mr. Putin talked about reviving discussions on “strategic stability,’’ the term of art among negotiators for talks that cover not just the number of nuclear weapons deployed on each side, but where they are based, how they are inspected, and steps to deter their use.

The last, tentative arms control talks were ended shortly before Russia’s full-scale invasion of Ukraine in 2022. Mr. Putin has argued since then that any talks on limiting nuclear arms should also cover the war in Ukraine. The Biden administration had refused to mix the two, fearing that Mr. Putin’s real goal was to trade limits on its nuclear arsenal for the territory he had captured in Ukraine and other concessions.

But Mr. Trump seems open to a broader negotiation, which is exactly what Mr. Putin would like, because it could enable him to make that trade-off.

It is unclear what, if any, long-term security guarantees Mr. Trump is willing to offer to President Volodymyr Zelensky, who he has insisted in recent days should have made a deal with Mr. Putin and avoided a devastating war.

Mr. Trump clearly wants to establish himself as a peacemaker: In his first term he suggested he deserved the Nobel Peace Prize, and bringing some kind of end to Europe’s biggest war since World War II would bolster his argument. He seems unconcerned about giving Ukraine a substantive role in the process, in contrast to former President Joe Biden, whose mantra was “Nothing about Ukraine without Ukraine.”

“For all these blustering exchanges, the thing Putin most wants to hear is that this is a deal Russia and the U.S. will strike by themselves,” said Stephen Sestanovich, a Russian and Eurasian studies expert at the Council on Foreign Relations, and a former State Department official.

Keith Kellogg, a retired general who, at 80, has been tasked by Mr. Trump to get the conversations going, insists that the key will be economics, not casualties. “When you look at Putin, you can’t just say, ‘Well, stop the killing,’ because candidly, that’s not their mentality,’’ he said on Fox News last week. Mr. Trump “approaches warfare differently: he looks at the economics as a piece of that warfare.” And he will focus, Mr. Kellogg insists, on limiting Russia’s oil revenues.

Mr. Putin, confident of his position on Ukraine’s battlefields despite Russia’s enormous casualties, has been trying to telegraph a wait-and-see approach to Mr. Trump. Russia’s war goals haven’t changed, he has said, and while it is ready for talks to end the fighting, it will only do so on its own terms.

Mr. Putin has strongly signaled that, at a minimum, he would demand to keep the roughly 20 percent of Ukraine that Russia now controls, as well as an agreement ruling out NATO membership for Ukraine and limiting the size of its military.

At the same time, Mr. Putin has made clear his eagerness to engage with Mr. Trump — and, more broadly, with the United States, after three years of diplomatic isolation by the Biden administration.

The Kremlin’s spokesman, Dmitri S. Peskov, has been telling journalists on a near-daily basis that Mr. Putin is ready to receive Mr. Trump’s call. “We’re waiting for signals,” he said Friday. “Everyone is ready.”

And Mr. Putin himself twice went out of his way last week to lavish praise on Mr. Trump — a proven method for winning Mr. Trump’s favor.

On Monday, Mr. Trump’s inauguration day, he held a televised meeting of Russia’s Security Council — an event that normally happens on Fridays and largely behind closed doors. He said Mr. Trump “showed courage” in surviving attempts on his life and had won “a convincing victory.”

On Friday, in a stage-managed moment, Mr. Putin stopped to answer a state television reporter’s question about Mr. Trump. The Kremlin promptly posted the video on its website.

“It is probably better for us to meet and, based on today’s realities, talk calmly about all areas that are of interest to both the U.S. and Russia,” Mr. Putin said. He brushed aside Mr. Trump’s sanctions threats, calling him “smart” and “pragmatic,” and spoke Mr. Trump’s language by saying the 2020 election had been “stolen” from him.

Like Mr. Trump, Mr. Putin has hinted at a desire to discuss a much broader set of issues with Mr. Trump than only the war in Ukraine. In his comments to state television on Friday, Mr. Putin said the Kremlin and the Trump administration could “jointly look for solutions to the key issues of today, including strategic stability and the economy.”

The “strategic stability” reference signaled potential interest in arms control talks, which the Kremlin briefly began with the Biden administration in 2021. “We discussed the range of arms control and nonproliferation issues, from AI in weapons to renewal of New START,’’ Wendy Sherman, the former deputy secretary of state, who conducted the talks for the U.S. side, said in an email. (New START is the arms control treaty that has been partly suspended by Russia, and expires in February 2026.)

Ms. Sherman noted that the talks were broken off ahead “of Putin’s horrific invasion.’’

Mr. Putin’s invitation for broad talks underscored what appears to be his continued optimism about Mr. Trump, despite Mr. Trump’s tough words about Russia last week and the fact that the president imposed a raft of new sanctions on Russia during his first term as president.

Mr. Trump also went after Volodymyr Zelensky, the Ukrainian president, last week, essentially blaming him for not striking an agreement with Mr. Putin that could have avoided the war.

“I could have made that deal so easily, and Zelensky decided that ‘I want to fight,’” Mr. Trump told the Fox Television host Sean Hannity.

He made clear he was not interested in Mr. Biden’s approach of supporting Ukraine for as long as necessary. But with his tough rhetoric against Mr. Putin last week, Mr. Trump may be trying to show he is not a pushover for the Russian leader, while preparing for the possibility that he cannot coax Mr. Putin into a deal that works for all sides.

“To keep Putin off balance, Trump has to show him a deal is possible only if it makes sense to Ukraine and our allies,” Mr. Sestanovich said.

Even as Mr. Putin welcomes talks with Mr. Trump, Russian officials aren’t backing away from their overall message about the United States as a malignant force — one sign of how the Kremlin is hedging its bets in case discussions with Mr. Trump do not go well.

Ms. Sherman, who has extensive experience negotiating with Russia, warns that if talks with Russia begin, the Trump administration should be ready. “Putin will want what he has always said he wanted: As much territory as possible, no Ukraine ever in NATO, no Western nuclear weapons in Europe that could target Russia.” Given that, she bets that actually negotiating a follow-on to the New START treaty “is likely low on his list.”