Israel and Hamas appeared close to an agreement to declare a cease-fire in Gaza and release hostages held there, the Qatari government, a key broker in the talks, said on Tuesday, raising hopes after more than 15 months of war for some respite in the fighting.

The latest round of negotiations follows repeated failed attempts to reach a breakthrough. But in recent weeks, officials familiar with the talks have voiced hope that a looming deadline was helping to close the gap: the end of President Joe Biden’s term and President-elect Donald J. Trump’s inauguration on Jan. 20.

Mediators had “managed to minimize a lot of the disagreements between both parties,” Majed al-Ansari, the Qatari Foreign Ministry spokesman, told reporters. The talks on Tuesday were focused on “the final details of reaching an agreement,” he said.

Officials in both the Israeli government and Hamas have suggested that they are ready to move forward if the other side signs off. On Monday, a Hamas official said a deal was possible in the coming days as long as Israel did not suddenly change its positions. On Tuesday, an Israeli official said Israel was ready to close the deal and was waiting for Hamas to make a decision.

But mediators, which also include Egypt and the United States, and other officials have warned that even substantial progress could be dashed at the last minute. Each of the previous rounds of negotiations over the past several months ultimately broke down in mutual recrimination.

“We believe that we are at the final stages, but until we have an announcement — there will be no announcement,” said Mr. al-Ansari, adding that there was no immediate timeline for signing a deal.

Mr. Trump has warned that there would be “ALL HELL TO PAY” unless the hostages were freed by the time he became president. Officials in the Biden administration had been pressing for a deal that would become part of the departing president’s legacy.

If Hamas and Israel conclude an agreement, it would bring some relief to Palestinians in Gaza, who have endured miserable conditions in displacement camps and relentless bombardments by Israel, and for the families of hostages abducted from Israel, who have worried for more than a year about the fate of their loved ones.

A framework agreement had been sent to both sides, said Mr. al-Ansari, who said the talks now centered on “outstanding details” about how the deal would be implemented.

In a statement, Hamas also said that the negotiations “had reached their final stages.” The Palestinian armed group’s leadership “hoped that this round of talks would end with a complete and clear agreement,” Hamas said.

Hamas officials negotiating in Doha must obtain the consent of the group’s remaining military commanders in Gaza for the emerging deal. Those commanders include Mohammad Sinwar, whose brother Yahya led the group before being killed by Israel in September. Communicating with them can be difficult, leading to delays.

It was still not clear whether Mr. Sinwar had conveyed to Hamas leaders in Doha whether he was on board with the proposed agreement.

The framework of the deal was heavily inspired by previous proposals discussed in May and July, said a diplomat familiar with the talks, who spoke on condition of anonymity to discuss the volatile negotiations. Those proposals detailed a three-stage cease-fire in which Israeli troops would gradually withdraw from Gaza, as Hamas released hostages in exchange for Palestinians jailed by Israel.

For over a year, international efforts have failed to end the war, which was ignited by the October 2023 Hamas-led attack that killed around 1,200 people. Another 250 were taken hostage to Gaza, according to the Israeli authorities. In response, Israel launched a military campaign against Hamas that destroyed large areas of the enclave and killed at least 45,000 people, according to Gaza health officials, who do not distinguish between civilians and combatants.

Around 105 hostages were released during a weeklong cease-fire in November 2023, the bodies of others were recovered by Israeli troops, and a handful were rescued alive. Roughly 98 hostages are now believed to remain in Gaza, around 36 of whom are presumed dead by the Israeli authorities.

During the first phase of the cease-fire — which would last roughly six weeks — Hamas will release 33 named hostages, most of whom Israel believes are alive, said an Israeli official, who requested anonymity to discuss the sensitive talks. Israel is willing to release hundreds of Palestinian prisoners in exchange, the official said, but the number depends on how many of the hostages are still alive.

Nonetheless, the Israeli official said that Israel and Hamas were very close to reaching an agreement. That optimism was echoed by Mr. Biden in a speech on Monday in which he declared that a cease-fire and hostage deal was on “the brink” of “finally coming to fruition.”

William J. Burns, the C.I.A. director, and Brett McGurk, a senior White House official, have crisscrossed the Middle East, pressing for a breakthrough in the talks. Steve Witkoff, Mr. Trump’s pick for Middle East envoy, has also made trips to Qatar and Israel, meeting with top officials there, including Prime Minister Benjamin Netanyahu of Israel on Saturday.

But while there is significant public pressure in Israel to reach a deal to free the hostages, many Israelis also fear that a cease-fire would effectively leave Hamas in power in Gaza, allowing its fighters to ultimately regroup and plan more attacks months or years down the road.

Two of Mr. Netanyahu’s hard-line coalition allies — Itamar Ben-Gvir and Bezalel Smotrich, have already denounced the proposed agreement as effective surrender to Hamas. The two far-right parties could threaten Mr. Netanyahu’s government if they withdrew from his ruling coalition in protest.

The agreement would likely still go through, as Israel’s parliamentary opposition has mostly committed to giving Mr. Netanyahu a safety net to secure a cease-fire and hostage deal. But it is unclear how long that would last, as it would leave Mr. Netanyahu’s political future dependent on rivals who have vowed to oust him.

In Gaza, Montaser Bahja, a displaced English teacher sheltering in Gaza City, said Palestinians were starting to feel hopeful that a deal could be imminent after more than a year of hunger and deprivation.

But even if both sides declared a cease-fire, many Gazans were frightened by a postwar future that was also far from certain, Mr. Bahja said. And even if Hamas’s deal secured the release of hundreds of Palestinian prisoners, few would see it as an achievement given the scale of the death and devastation in Gaza, he added.

“Everything is up in the air,” he said. “At this point, people just want to it end.”

Ukraine carried out “massive” drone strikes on several regions of Russia overnight, local officials there said on Tuesday, in what appeared to be one of the largest recent assaults in Kyiv’s campaign to cripple Russia’s war machine on its home turf.

The attacks, mostly in southwestern Russia, were the latest in a series that have demonstrated Ukraine’s ability to strike deep inside the country, even as Kyiv’s forces face setbacks on their own territory.

Blasts were reported in the border region of Bryansk, and drones also targeted regions well beyond it like Saratov and Tula in western Russia, officials in those areas said.

Russia’s Defense Ministry said on Tuesday that Ukraine had launched more than 140 drones, along with U.S.-made long-range missiles known as ATACMS and Storm Shadow missiles provided by Britain.

“These actions of the Kyiv regime, supported by Western curators, will not go unanswered,” the ministry said in a statement.

Ukraine’s military did not immediately comment on the strikes. But Ukraine’s Center for Strategic Communications and Information Security, a government agency, said seven regions of Russia had been attacked and that the targets included military production factories as well as oil and gas facilities.

The threat forced at least six cities to restrict their airspace Tuesday morning, according to a statement from Russia’s Federal Aviation Agency. Those included the cities of Saratov and Engels, which were attacked overnight.

Two industrial plants sustained damage, Roman V. Busargin, the governor of the Saratov region, wrote on Telegram. “Today Saratov and Engels were subjected to a massive UAV attack,” he said, using another name for drones. “Air defenses eliminated a large number of targets.”

It was the second time in a week that Engels, which is the site of an airfield for some of Russia’s long-range, nuclear-capable bombers, has been attacked. Emergency crews only recently extinguished a large fire sparked by a strike on Jan. 8.

Mr. Busargin said that schools would be closed and classes in Engels and Saratov would be held remotely on Tuesday.

The authorities in the Tula region of western Russia similarly confirmed a drone attack that they also described as massive. Dmitry V. Milyaev, the regional governor, said air defenses shot down 16 drones and that falling debris had damaged some cars and buildings. There were no casualties.

And local news media in Kazan, the capital city of the republic of Tatarstan in southwestern Russia, reported that a tanker at a liquefied natural gas base was struck, igniting a large fire.

The head of the republic, Rustam N. Minnikhanov, wrote on Telegram that firefighters had put out the blaze and that there were no casualties or “significant damage.”

In the border region of Bryansk, which has come under more regular attack, powerful explosions were reported. The Russian independent media organization Astra said that a chemical plant had been struck; the report could not be independently verified.

Russia’s Defense Ministry said it shot down 31 drones, six ATACMS and six Storm Shadow missiles targeting Bryansk.

From the start, Kyiv’s strikes inside Russia have aimed at limiting Moscow’s ability to attack Ukrainian cities. But in recent weeks they have taken on added weight in an attempt to project strength before President-elect Donald J. Trump’s inauguration next week, amid concerns he might temper U.S. support for Ukraine.

Mr. Trump has vowed to end the war swiftly. While he has not said how, many in Ukraine fear that he could make concessions to President Vladimir V. Putin of Russia that have been red lines for Kyiv.

Ahead of Mr. Trump’s inauguration on Monday, the Russian military has also been putting on a show of force. While Kyiv’s drones were targeting regions of Russia overnight, Moscow’s forces were carrying out an aerial assault on Ukraine that put much of the country under air-raid alerts.

Ukraine’s Air Force said on Tuesday morning that nearly 80 drones were involved in the attack but that it managed to shoot down 60. Apartment buildings and cars sustained damage from downed drones in several regions, it added, but there were no casualties.

Nataliia Novosolova and Nataliya Vasilyeva contributed reporting.

He is a deeply unpopular figure in Britain, according to opinion polls, and his social media channel has lost users in the country since he took it over in October 2022. Yet when Elon Musk put Britain in his cross hairs on X in recent weeks, pounding the political establishment over a decade-old child sex abuse scandal, he instantly catapulted the issue to the top of the news agenda.

Mr. Musk’s success is rooted in two obvious factors: his mammoth fortune and his alliance with the incoming president of the United States, Donald J. Trump. But it also reflects a British political and media establishment that is divided and deeply in flux, all of which has made Britain easy pickings for an outside influencer with vast resources and a single-minded mission to disrupt.

Britain’s right-leaning newspapers have picked up and amplified Mr. Musk’s call for a new national investigation of young girls who were sexually exploited in several towns in the 2000s, including in Rotherham, where an estimated 1,400 girls were exploited by “grooming gangs” composed largely of British Pakistani men.

The leader of the opposition Conservative Party, Kemi Badenoch, threw her support behind a new national investigation as well, even though the previous Conservative government did not pursue one. A former Tory leader, Boris Johnson, once belittled those inquiries, saying, an “awful lot of money and an awful lot of police time now goes into these historic offenses and all this malarkey.”

The Labour government, which was vaulted into power with a landslide majority in July, has so far rejected Tory calls for another investigation, saying its priority is to implement the recommendations from a previous seven-year-long national investigation, including tightening requirements to report child abuse and collect better data on cases.

The government’s minister for safeguarding and violence against women and girls, Jess Phillips, had earlier pushed back privately on calls for a fresh national inquiry, arguing that a local inquiry would be more effective, for which she was labeled a “rape genocide apologist” in a post by Mr. Musk.

But even as Prime Minister Keir Starmer defended Ms. Phillips, the drumbeat of angry Musk posts forced him to retreat from his claim that Ms. Badenoch and others were calling for an investigation simply “because they want to jump on a bandwagon of the far right.” Now, he says, he has an open mind about it.

The crimes committed against young girls in these cases were so appalling that the resurfacing of them, even years later, would have drawn a strong public reaction, regardless of how they came to light.

But as Sunder Katwala, the director of British Future, a research organization in London, said, “Elon Musk’s political posting has been enormously influential in driving both the political and media conversation in the U.K.”

Based strictly on the numbers, Mr. Katwala said, that makes little sense: Mr. Musk is viewed favorably by only 20 percent of people in Britain, according to the market research firm YouGov, while 71 percent view him unfavorably. The reach of X fell by eight percent from May 2023 to May 2024, according to an annual report on online usage by Ofcom, which regulates communications in Britain.

“There has been a recent tradition of lecturing the liberal left to remember that Twitter is not Britain,” Mr. Katwala said. “But the right-wing ecosystem seems inclined to forget that X is not Britain either.”

Still, he said, X has managed to hang on to its followers among the politicians and journalists who work in Westminster and who set the political news agenda. They have proven a rapt audience for Mr. Musk’s increasingly strident, often erroneous, posts about Britain and his false claims about Mr. Starmer’s supposed complicity in a scheme to cover up child sex abuse, dating back to his days as Britain’s director of prosecutions. In fact Mr. Starmer’s office brought the first case against a grooming gang and drafted new guidelines for the mandatory reporting of child sex offenses.

While many of Mr. Musk’s posts, particularly those on grooming gangs, originated in the ecosystem of far-right bloggers and activists, they are also tempting to mainstream politicians in search of a cudgel to use against their opponents. And they appeal to editors and broadcasters looking for a good story.

“The British press and the broadcasters, to a degree, fell all over themselves to give Elon Musk publicity,” said David Yelland, a former editor of Rupert Murdoch’s tabloid, The Sun. “In the print press, they did it because they are extremely hostile to Keir Starmer. This is plain old Fleet Street bias.”

Claire Enders, a London-based media researcher and founder of Enders Analysis, likened Mr. Musk to Mr. Murdoch, the insurgent media baron from Australia who upended the London newspaper industry in the 1970s. “We just have a new Murdoch,” she said. “He’s American, he’s a multibillionaire, and he’s close to Trump.”

Mr. Musk, however, is not interested in taking over the British press so much as discrediting it. He claims the news media was complicit in a coverup of abuses against young girls. The truth is, British newspapers across the political spectrum did cover these crimes, if not immediately, then energetically, as the scale of the abuses became apparent in the late 2000s and early 2010s. The Times of London published a major investigation of the scandal, and the slow response to it by the police, in 2011.

“It’s been on the front page of every paper and led the 6 o’clock news for years,” said Raheem Kassam, who covered the scandal as editor of the British outpost of the right-wing news outlet, Breitbart News. “The idea that there is a media blackout on this, and we needed Elon Musk to uncover it, is nonsense.”

And yet even The Times of London, which is also owned by Mr. Murdoch, published its own demand for a new inquiry on its front page on Jan. 6. It pointed to the hundreds of articles it had published on the scandal but added that a newspaper “can only go so far.”

No British media outlet has revived the grooming scandal with the zeal of GB News, a hard-right cable news channel that went on the air in 2021, a decade after The Times’s investigation into grooming gangs. Charlie Peters, an investigative reporter, broke the story that Ms. Phillips had rejected a request for a national inquiry into child sexual abuse in Oldham, a town near Manchester.

Since then, GB News has interviewed relatives of the victims of abuse across Britain. Nigel Farage, the leader of Reform U.K., an anti-immigrant party, has praised Mr. Peters, saying he had “really reignited this story” and demonstrated that “these barbarities have taken place in at least 50 towns.” Mr. Farage has vowed that Reform will carry out its own investigation if the government does not.

But Mr. Farage is also trying to extricate himself from his own spat with Mr. Musk — this one over Mr. Farage’s refusal to echo Mr. Musk’s demand that Tommy Robinson, a far-right agitator with multiple criminal convictions, be released from prison. Mr. Musk posted that Reform needed to find a new leader because Mr. Farage “doesn’t have what it takes.”

The cumulative effect of Mr. Musk’s inflammatory posts has been to energize Britain’s populist right. Even Mr. Farage’s rift with Mr. Musk may ultimately play to his benefit, giving him credibility with those who revile Mr. Robinson. The Labour government, meanwhile, is struggling to regain control over its agenda, while critics say the Conservatives embraced Mr. Musk largely to stay relevant.

“He brings the two tactical nuclear weapons of modern politics: unlimited cash and a social media platform,” said Stephen K. Bannon, a former adviser to Mr. Trump who has clashed with Mr. Musk on visas for skilled workers in the United States. “There’s not a government in Europe that can withstand this guy’s onslaught.”

Mr. Kassam, a former chief of staff to Mr. Farage who now works with Mr. Bannon in the United States, said he expected Mr. Farage and Mr. Musk to mend fences soon. Aside from their differences over Mr. Robinson, he said, their political philosophies were probably closer than those of Mr. Musk and Mr. Trump.

“Elon tries to break things and see where they fall,” Mr. Kassam said. “He’s treating the United Kingdom like one of his companies.”

Greenland is ready to talk.

Responding on Monday to the diplomatic earthquake set off last week by President-elect Donald J. Trump, who mused about taking over the gigantic island in the Arctic Ocean, Greenland’s prime minister said the territory would like to work more closely with the United States on defense and natural resources.

“The reality is we are going to work with the U.S. — yesterday, today and tomorrow,” Prime Minister Múte Egede said at a news conference in Nuuk, Greenland’s tiny, icebound capital.

But he was firm: Greenlanders did not want to become Americans.

“We have to be very smart on how we act,” he said, adding, “The power struggles between the superpowers are rising and are now knocking on our door.”

Mr. Trump refused to rule out using economic or military force to wrest back the Panama Canal and to take Greenland, a semiautonomous territory of Denmark that he suggested buying during his first term in office. Then, as now, Greenland and Denmark said the island was not for sale. Panama’s leaders, too, rejected the threat.

Mr. Egede said on Monday that “all of us were shocked” by Mr. Trump’s words, which were accentuated by a surprising and somewhat mysterious visit by the president-elect’s eldest son, Donald Trump Jr., to the island on the same day.

The younger Mr. Trump did a lightning-fast sightseeing tour, saying he was on private business, and since then, headlines around the world have blared Greenland.

Most of Greenland’s territory is covered in ice, only about 56,000 people live here and, until recently, the island was best known for its icebergs and polar bears. As climate change melts the Arctic ice, this region has been quietly falling into the cross hairs of the world’s powers.

The United States, Russia, European countries, China and others have been eyeing the Arctic’s shipping lanes and the extensive mineral resources that are no longer considered out of reach.

The island has been tied to Denmark for centuries, first as a colony and now as a separate territory that has achieved a large degree of autonomy in recent years. Denmark still controls the island’s foreign affairs and defense policy.

But the surge of interest by international powers dovetails with Greenland’s quest to gain independence, and that itch has only grown stronger. At the same time, many people here are reluctant to completely cut ties with Denmark because of the hundreds of millions of dollars in subsidies that Denmark provides each year.

In Nuuk, which was a sunny zero degrees Fahrenheit on Monday afternoon, many people were eagerly waiting to hear what the prime minister was going to say.

“Whatever happens, there’s no turning back,” said Aviaq Kleist, the owner of a cafe in the Nuuk Center, the city’s biggest mall, with a couple dozen shops. She joked that maybe Mr. Egede would suddenly declare independence.

Mr. Egede did not — he danced around the question, saying the country had been steadily working toward the goal but that “different parties have different views.” (There’s also a clear independence process that involves a referendum, should it come to that.)

The prime minister also expressed relief at comments that JD Vance, the incoming vice president, made on a Fox News show this weekend. Though Mr. Vance didn’t exactly rule out military force, saying, “We don’t have to use military force” because “we already have troops in Greenland,” his tone was upbeat as he spoke about Greenland’s “incredible natural resources” and “a deal to be made.”

The United States has been interested in Greenland for years. During World War II, it established bases here, and after the war, it tried to buy Greenland from Denmark, which refused. Today, the American military runs the Pituffik Space Base, which specializes in missile defense, at the northern end of the island.

In Nuuk on Monday, people seemed to be on the same page as the prime minister, expressing a mix of hope and caution. Several said they did not want to be swallowed by the United States. But they did want a stronger partnership with America.

“What we really need is more cooperation and trade,” said Nielseeraq Berthelsen, a fisherman. He was working at an ice-encrusted seafood market, selling hunks of whale skin and bright red seal meat.

He said that he was walking through another mall last week when someone approached him out of the blue and invited him to a special dinner.

Next thing he knew, he said, he was shaking the younger Mr. Trump’s hand.

“He had a lot of enthusiasm,” said Mr. Berthelsen, who was standing in air so cold that his eyes watered as he talked. “He had good energy.”

Ivik Kristiansen contributed reporting.