But Mr. Netanyahu’s statement on Sunday night dampened those expectations, since it appeared to lessen the chances of a compromise with Hamas over the length and permanence of the cease-fire.

“Any deal will allow Israel to resume fighting until all of the objectives of the war have been achieved,” his statement said, reiterating his long-held position that the war must continue until Israel has destroyed Hamas’s military and governing abilities.

To Mr. Netanyahu’s critics, his intervention — at such a sensitive moment in the negotiations — risked derailing the efforts to secure the release of roughly 120 hostages Israeli officials say are still held in Gaza, both dead and alive, after being captured by Hamas and its allies at the start of the war in October.

“We’re at a critical moment in the negotiations. The lives of the hostages depend on them. Why issue such provocative statements?” Yair Lapid, the opposition leader, wrote on social media. “How does that contribute to the process?”

Analysts said that Mr. Netanyahu’s intervention showed how he is trying to balance the effort to free the hostages with his desire to hold together his coalition of ultranationalist and ultrareligious political parties.

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.

“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. It’s that simple.”

Others more sympathetic to Mr. Netanyahu say he may 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, said Nadav Shtrauchler, a former strategist for the prime minister. “The efforts of the military in Gaza may help him get more from Hamas,” Mr. Shtrauchler said.

Additionally, Mr. Netanyahu may be trying to stave off the collapse of his coalition 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, Mr. Shtrauchler said.

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

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.