The Israeli military on Monday said it was asking tens of thousands of Gazans sheltering in eastern Rafah to temporarily evacuate to what it described as a humanitarian zone, a sign that Israel was inching closer to invading the city in defiance of international pressure.

By 9 a.m. local time, the military had begun dropping leaflets in eastern Rafah ordering people to evacuate. The Israeli military said it would also use text messages, phone calls and broadcasts in Arabic to warn residents of Rafah to leave.

Since the start of Israel’s ground offensive in Gaza last year, about a million people have fled to Rafah, the southernmost city in the enclave, where they have been living in dire conditions. Israel has told civilians in many parts of Gaza to evacuate from their homes for safety since the start of the war. But in many instances the places Israel said would be safe for Gazans were also targeted by Israeli airstrikes.

Israel’s closest allies, including the United States, have been urging it to not send troops into Rafah, saying such an operation would take a heavy toll on civilians. But Prime Minister Benjamin Netanyahu has repeatedly rejected those calls, saying Israel needs to defend itself and eliminate Hamas, which attacked Israel on Oct. 7.

Salama Marouf, the head of the Hamas-run Gaza government media office, said in a statement that the evacuation order showed that Israel “went into truce negotiations deceptively without abandoning the idea of a broad aggression against Rafah.” He said the announcement was “a real test of the seriousness” of the countries that had warned against an invasion of the city.

Mr. Netanyahu’s stance on Rafah has complicated cease-fire talks and raised concerns about the hostages Hamas is still holding. An agreement has remained elusive, as Mr. Netanyahu has indicated that he is open to only a temporary halt in fighting while Hamas has sought a permanent one. On Sunday, Mr. Netanyahu repeated his promises to destroy Hamas.

“If Israel is forced to stand alone, Israel will stand alone,” Mr. Netanyahu said in English, in a speech marking Israel’s Holocaust Remembrance Day. “But we know we are not alone because countless decent people around the world support our just cause. And I say to you, we will defeat our genocidal enemies. Never again is now!”

About two weeks ago, the Israeli authorities said that before they moved on Rafah, they would expand a humanitarian zone in nearby Al-Mawasi where civilians could shelter. On Monday, the Israeli military said that it had done so, and that the zone had field hospitals, tents and larger supplies of food, water and medicines.

The military is not calling for a “wide-scale evacuation of Rafah,” Nadav Shoshani, a spokesman, said on Monday. “This is a very specific scoped operation at the moment to move people out of harm’s way.”

On Monday, the Israeli military, known formally as the Israel Defense Forces, seemed to foreshadow its next move in Rafah.

“The I.D.F. will continue pursuing Hamas everywhere in Gaza until all the hostages that they’re holding in captivity are back home,” it said in a statement.

UNRWA, the U.N. agency that aids Palestinian refugees, said on Monday that it would not evacuate its staff from Rafah and would continue to provide humanitarian aid to those who have taken refuge there.

“An Israeli military offensive will lead to an additional layer of an already unbearable tragedy for the people in Gaza,” Philippe Lazzarini, the agency’s commissioner general, said on social media.

Prime Minister Benjamin Netanyahu of Israel on Sunday rejected international pressure to rein in its military campaign in Gaza and, speaking at a Holocaust memorial, asserted Israel’s right to fight its “genocidal enemies.”

Nearly seven months into the war, Mr. Netanyahu has been steadfast in his goal of destroying Hamas. This, and Mr. Netanyahu’s insistence on sending troops into Rafah, the southernmost city in the Gaza Strip, has complicated efforts to end the fighting and raised concerns about the future of the hostages held by Hamas.

But Mr. Netanyahu has remained defiant.

On Sunday, he spoke at Yad Vashem, Israel’s Holocaust memorial in Jerusalem, to mark the national Holocaust remembrance day. Hamas’s Oct. 7 attack, he said, was not a “Holocaust” — not because Hamas did not have the intention to destroy Israel but because of its inability to do so. About 1,200 people were killed and more than 200 were taken hostage that day, Israeli authorities say. Hamas’s intention, Mr. Netanyahu said, was the same as that of the Nazis.

In his speech, which lasted for about 15 minutes and was largely in Hebrew, Mr. Netanyahu rejected accusations that Israel was committing genocide in the Gaza Strip. Since the beginning of the war, Gazan authorities say Israeli troops have killed more than 34,000 people, many of them women and children, though the statistics do not differentiate between civilians and combatants.

Mr. Netanyahu said that Israel’s military does everything it can to avoid harming civilians and that it has allowed aid to flow through to Gaza to avoid a humanitarian crisis. A United Nations official recently said that parts of Gaza are experiencing “full-blown famine.”

Mr. Netanyahu made a point to say a few words in English that were aimed at the international community. He invoked the Holocaust in asserting Israel’s right to defend itself, with or without international support.

“If Israel is forced to stand alone, Israel will stand alone,” he said. “But we know we are not alone because countless decent people around the world support our just cause. And I say to you, we will defeat our genocidal enemies. Never again is now!”

On Monday morning after his speech, the Israeli military gave the strongest signal yet that it was going to invade Rafah as it asked tens of thousands of Gazans to evacuate from the city.

The latest round of negotiations between Israel and Hamas hit an impasse on Sunday as mediators struggled to bridge remaining gaps and a Hamas delegation departed the talks in Cairo, according to two senior Hamas officials and two other officials familiar with the talks. An Israeli official also confirmed the negotiations had stalled and described them as being in “crisis.”

For months, the negotiations aimed at achieving a cease-fire and a release of hostages have made little progress, but signs the two sides were coming closer to an agreement appeared over the last week. Israel backed off some of its long-held demands and a top Hamas official said the group was studying the latest Israeli offer with a “positive spirit.”

But the setback over the weekend meant Palestinians living in miserable conditions in Gaza would not experience an imminent reprieve and the families of hostages held by militants would have to wait longer for the freedom of their loved ones.

The main obstacle in the talks was the duration of a cease-fire, with Hamas demanding it be permanent and Prime Minister Benjamin Netanyahu of Israel expressing openness to only a temporary halt in the fighting.

Hamas blamed the lack of progress on Mr. Netanyahu, who vowed again in recent days that the Israeli army will invade Rafah, the southernmost town in the Gaza Strip, with or without an agreement.

“We were very close, but Netanyahu’s narrow-mindedness aborted an agreement,” Mousa Abu Marzouk, a senior Hamas official, said in a phone interview.

Mr. Netanyahu has for weeks declared his intention to stage a ground offensive aimed at Rafah, where about a million Palestinians have been sheltering. The Biden administration has been pressing Israel to refrain from undertaking a major operation in the city.

On Sunday, Hamas fired roughly 10 rockets from the area of the Rafah border crossing, killing three soldiers near the Kerem Shalom crossing, according to Israel’s military. Rocket attacks by Hamas have been relatively rare in recent months, and Israel said it had responded with airstrikes targeting the site of the launches.

The Israeli official who spoke on the condition of anonymity said that Israel and Hamas were closer to a deal a couple of days ago, but that Mr. Netanyahu’s statements about Rafah had compelled Hamas to harden its demands in an attempt to ensure that Israeli forces won’t enter the city. Hamas, the official said, was now seeking further guarantees that Israel would not implement only part of an agreement, and then resume fighting.

The official lamented that Hamas and Israel had shifted gears to playing a “blame game.”

Two U.S. officials, who spoke on the condition of anonymity, pushed back on the notion the talks were at an impasse, suggesting that parties were still reviewing details of the most recent proposals.

Mr. Netanyahu and the United States have been contending that Hamas was holding up an agreement. On Sunday, Mr. Netanyahu said he would not agree to the complete withdrawal of Israeli forces from Gaza and an end to the war. Countenancing such demands, he said, would allow Hamas to re-establish control over Gaza, rebuild their military capabilities, and threaten communities throughout Israel.

“It is Hamas that is holding up the release of our hostages,” he said. “We are working in every possible way to free the hostages; this is our top priority.”

An Israeli delegation never made it to Cairo for the latest round of talks. The Israeli official said that Israel had sought a written response to its latest proposal from Hamas before dispatching a delegation, but that the group never conveyed one.

Mr. Abu Marzouk said Hamas had wanted Israel to be present at the talks in Cairo, where they could have worked through mediators to clarify “vague” points in the latest Israeli offer, including on the duration of a cease-fire.

“The cease-fire needs to be permanent and fixed,” he said.

Mr. Abu Marzouk was the only one of the officials who spoke about the talks to allow the use of his name. The others spoke on the condition of anonymity to discuss the sensitive subject or because they were not authorized to speak to the press.

Hamas, Mr. Abu Marzouk said, thought that Mr. Netanyahu wanted an agreement that would permit Israel to invade Rafah after its hostages are released.

“This is Netanyahu’s plan,” he said.

A technical team from the Qatari foreign ministry also left the Egyptian capital on Sunday, two officials briefed on the talks said. Bill Burns, the director of the Central Intelligence Agency, met with Qatari Prime Minister Mohammed bin Abdulrahman Al Thani in Doha on Sunday to discuss getting the talks back on track, one of the officials said.

On Monday, Hamas’s political leadership will convene in Doha to discuss what unfolded in Cairo over the past two days, but the group intended to continue participating in negotiations with “positivity,” said one of the senior Hamas officials, who spoke on the condition of anonymity.

A report in Al-Qahera News, an Egyptian state-owned television channel, said that a Hamas delegation would return to Cairo on Tuesday, but the senior Hamas official said that the group hadn’t made a decision yet.