On Thursday, the Hamas leader, Ismail Haniyeh, told an Egyptian official that his group was examining the proposal. A Hamas delegation will go to Egypt soon to “complete the ongoing discussions” for a deal that “realizes our people’s demands and ends the aggression,” according to a statement from the group.

Less than a day earlier, a Hamas spokesman, Osama Hamdan, had said on Lebanese television that “our position on the current negotiating paper is negative.” But the Hamas press office later said his comment was not an outright rejection. Some changes would need to be made for Hamas to agree, the office said, without elaborating.

In Israel, the war cabinet was scheduled to meet on Thursday evening to discuss the cease-fire negotiations and a planned Israeli invasion of Rafah, the southernmost city in Gaza, where around a million people have been sheltering, according to an Israeli official who requested anonymity because they were not authorized to communicate with the media on the matter.

On a visit to Israel on Wednesday, Secretary of State Antony J. Blinken put the onus squarely on Hamas to accept the Israeli proposal. “We are determined to get a cease-fire that brings the hostages home and to get it now, and the only reason that that wouldn’t be achieved is because of Hamas,” he said.

However, the Israeli opposition leader, Yair Lapid, said that Prime Minister Benjamin Netanyahu had “no political excuse” not to make a deal quickly.

The complex negotiations have dragged on for months, with each bargaining piece moved also shifting several others. Complicating matters further is that Israel and the United States do not talk directly with Hamas, which they consider a terrorist organization, but communicate through officials of Qatar and Egypt who act as intermediaries.

A seemingly intractable sticking point is Israel’s planned ground offensive into Rafah. “If the enemy carries out the Rafah operation, negotiations will stop,” Mr. Hamdan said on Wednesday. “The resistance does not negotiate under fire.”

The Biden administration has pressed the Israeli government hard to abandon the idea of a major invasion of the city, and to rely instead on surgical operations to kill or capture Hamas leaders and fighters.

But Israeli officials have said, consistently and emphatically, that the offensive will take place. Far-right parties in Mr. Netanyahu’s coalition have hinted at leaving it if he calls off the offensive, which could cause the government to collapse and force new elections.

“We will enter Rafah and we will eliminate the Hamas battalions there — with or without a deal — in order to achieve the total victory,” Mr. Netanyahu said in a statement released on Tuesday.

Hamas has insisted that any agreement would eventually lead to a permanent cease-fire, not just a temporary halt in the fighting — a stance that Israel has rejected as a Hamas play for time to re-establish itself as a governing and military force. The Biden administration has held out hope that a six-week pause in the war could be the first step toward a lasting end to the fighting.

Israel this week softened some of its positions. It agreed to allow Palestinians to return to northern Gaza en masse in the first phase of a cease-fire. Israel had previously insisted on screening returnees and limiting their flow.

Israel also backed away from its demand that Hamas release 40 hostages — female civilians and soldiers, and those who ill or aged — after Hamas indicated that it did not have 40 living hostages in those categories. The latest proposal lowers the figure to 33. The number of Palestinians Israel is offering to free in exchange is unclear.

In the Hamas-led Oct. 7 assault on Israel, about 250 people were kidnapped and taken back to Gaza, according to the Israeli government. More than 100 were released in a weeklong cease-fire in November, and Israeli officials say they believe that more than 30 — possibly many more — are dead.

The Oct. 7 attacks killed some 1,200 people, Israel has said. Gazan health officials say that Israel’s subsequent bombing and invasion have killed more than 34,000 people, and injured far more.