Mr. Netanyahu’s insistence on being able to resume fighting drew widespread criticism on Monday in Israel, where there is growing support for a cease-fire deal that would involve the release of at least some of the Israeli hostages still held in Gaza.

Mr. Netanyahu’s governing coalition depends on support of ultranationalist leaders opposed to a permanent truce, and the backlash on Monday revived accusations that he was putting his personal interests above the nation’s.

Mr. Netanyahu insists that Hamas’s total defeat is in Israel’s strategic interest. But others say that the hostages’ freedom is a higher priority and that the prime minister’s main motivation is to avoid the collapse of his government.

The claims exemplify a wider dispute about Mr. Netanyahu, whose decision in 2020 to remain in politics, despite standing trial for corruption, worsened deep splits in Israeli society and prompted years of political instability.

The country’s military leadership believes that a cease-fire deal would be the swiftest way of releasing some 120 Israelis, some alive and some dead, who remain in Gaza. Recent polling also suggests that a majority of Israelis see the return of the hostages as a higher priority than continuing the battle against Hamas in Gaza.

Still, negotiations over a deal continued on Monday in Cairo, where Ronen Bar, the head of the Shin Bet, Israel’s domestic intelligence agency, gathered for talks mediated by the Egyptian government. More discussions are scheduled for later this week in Qatar, another mediator between Israel and Hamas. Some officials and analysts said that the comments from both Mr. Netanyahu and Hamas could be construed as an attempt to drive a hard bargain, rather than a rejection of the negotiating process.

Along with reaching a compromise over the length of the cease-fire, the sides need to agree on the number and identity of the Palestinian prisoners to be exchanged for the hostages. They also need to agree about the extent to which Israeli troops should withdraw from Gaza; Hamas seeks a total withdrawal, while Israel hopes to retain control over some parts of the territory that it has captured.

After months of failed negotiations, hopes for a deal were revived last week amid reports that Hamas had become more flexible on critical points, prompting Mr. Netanyahu to send negotiators to Qatar.

But Mr. Netanyahu’s grip on power relies on the support of two far-right parties opposed to any agreement that would leave Hamas in power in Gaza. Critics say this has made him wary of committing to a hostage-release deal that could lead to the collapse of his coalition and prompt early elections that polling suggests he would lose. Mr. Netanyahu is currently standing trial on charges of corruption, accusations he denies, and his political future would be at stake if he lost power for the third time in his career.

“The simple truth is as follows: Benjamin Netanyahu doesn’t want a hostage deal,” Ben Caspit, a biographer and prominent critic of the prime minister, wrote on social media. “He might be willing to get the hostages back, but not at the expense of his coalition’s well-being.”

An influential minister in Mr. Netanyahu’s coalition, Bezalel Smotrich, underscored the coalition’s fragility on Monday by hinting on social media that his party could leave the coalition if the prime minister struck a deal that kept Hamas in power in Gaza.

“We will not be part of a deal to surrender to Hamas,” said Mr. Smotrich, a far-right firebrand whose party holds the balance of power in Mr. Netanyahu’s ruling coalition.

“This deal is a defeat and humiliation for Israel,” Mr. Smotrich added.

Some analysts believe Mr. Netanyahu may not personally be opposed to a deal but wants to maximize its chances of success by delaying it until the end of July, when Parliament goes on recess.

Without a sitting Parliament, lawmakers would find it far harder to bring down the government, giving Mr. Netanyahu more room to strike a deal that his coalition partners might resist, according to Nadav Shtrauchler, a former strategist for the prime minister.

“He’s trying to create room for maneuver — and for that, he needs time,” Mr. Shtrauchler said.

Mr. Netanyahu may also be using hardball negotiating tactics in order to force bigger compromises from Hamas. With each passing day, Israel’s military operation in the southern Gaza city of Rafah further weakens Hamas’s position there, Mr. Shtrauchler said.

“The efforts of the military in Gaza may help him get more from Hamas,” Mr. Shtrauchler said.

However, the leader of the opposition, Yair Lapid, working on the assumption that Mr. Netanyahu’s resistance to a swift deal is mainly political in nature, on Monday offered to help keep the prime minister in power if the government collapsed over a hostage deal.

“It is not true that he has to choose between the life of the hostages and the continuation of his tenure as prime minister,” Mr. Lapid said in a speech. “I promised him a safety net, and I will keep that promise.”

Mr. Netanyahu did not immediately respond to Mr. Lapid’s offer, but analysts and allies of the prime minister said he was unlikely to accept it because he does not trust Mr. Lapid’s intentions.

“Lapid will give him a parachute for this specific deal, but 24 hours later he will vanish,” Mr. Shtrauchler said.

“It’s not something that Netanyahu can consider reliable,” Mr. Shtrauchler added.

Gabby Sobelman contributed reporting from Rehovot, Israel.

It added that the area it had moved into included the headquarters of the main United Nations agency that assists Palestinian refugees, known as UNRWA. Juliette Touma, the agency’s communications director, said it did not have any information about the military’s actions, but she noted that the agency had left its headquarters in October.

Israeli forces have repeatedly found themselves returning to parts of Gaza that they had previously left, especially in the north, which they invaded in October, as Hamas regroups amid the chaos of the nine-month war. The fighting has flared even as Prime Minister Benjamin Netanyahu has spoken of a new, less intense phase of the war.

The operation shows that Israel is still struggling to achieve one of its stated objectives in the war: wiping out Hamas, which organized and led the Oct. 7 attacks on Israel that set off the war in Gaza.

The military said it had warned civilians about its activity and opened a “defined route” for their evacuation. Wafa, the Palestinian Authority’s official news agency, reported “a massive exodus of thousands” of Palestinians from the area toward the northwestern neighborhoods of Gaza City.

The agency said Israeli military vehicles had entered areas in southwestern Gaza City, including the Tal al-Hawa neighborhood and the southern outskirts of the upscale Al-Rimal neighborhood. It added that residents reported heavy aircraft and artillery fire that killed and wounded dozens of people.

Ahmed Saleh, 44, who lives near Al-Rimal with 13 family members, said in a phone interview that the strikes began suddenly, “all kinds of strikes — tank shelling, artillery and aircraft.” He said they began receiving calls and text messages from the Israeli military telling them to evacuate, but only after the bombardment had begun.

Mr. Saleh said he managed to grab a bag with important documents and some clothes before fleeing with his family, including his 70-year-old mother, whose wheelchair broke on the way because of the bombed-out roads.

After borrowing another wheelchair, the family walked for another hour amid continued strikes, until they arrived at the house of Mr. Saleh’s sister outside the evacuation zone.

Mr. Saleh said that while fleeing his neighborhood, he saw “dozens of people” who had been killed or injured. He told people along the way to move west, “but they had to go east to collect their stuff and rescue their families,” he said. Many were caught in the bombardment, and “no one could reach them for help,” he said.

The Palestinian Red Crescent said in a statement that its emergency and rescue teams had evacuated at least 30 wounded people from a hospital in the Rimal neighborhood to the Indonesian Hospital, which is outside the evacuation zone.