Hamas says it has received some ideas from the US through mediators on how to reach a Gaza ceasefire agreement.
The Palestinian armed group said it was discussing how to develop the ideas, noting that it was ready to release its 48 remaining hostages in Gaza in return for a clear declaration ending the war.
A Palestinian official told the BBC the US plan would see the hostages freed in the first 48 hours of a 60-day truce in exchange for Palestinian prisoners in Israeli jails and good-faith negotiations on a permanent ceasefire.
It came after US President Donald Trump gave Hamas what he called a last warning to agree a deal. He said Israel had accepted his terms, without giving details.
Israel's Foreign Minister, Gideon Saar, said it was ready to agree a deal ending the war that would include the release of all the hostages - only 20 of whom are believed to be alive - and the disarmament of Hamas.
Prime Minister Benjamin Netanyahu meanwhile warned the approximately one million Palestinians living in famine-stricken Gaza City to evacuate immediately, as the Israeli military stepped up an offensive to conquer it.
In the last two days we brought down 50 terrorist high-rises, and this is only the beginning of the ground operation in Gaza City, he said in a video. I say to the residents of Gaza City: you have been warned, get out of there.
Hospitals said Israeli attacks killed at least 40 people across Gaza on Monday, including 25 in Gaza City and elsewhere in the north.
The Israeli military said four soldiers were killed in combat in northern Gaza.
On Sunday evening, Hamas put out a statement saying it had received through the mediators some ideas from the American side aimed at reaching a ceasefire agreement and that it welcomed the initiative.
The movement is in constant contact with the intermediaries to develop these ideas into a comprehensive agreement that meets the needs of our people, it added.
Earlier in the day, US President Donald Trump had written on Truth Social: Everyone wants the Hostages HOME. Everyone wants this War to end! The Israelis have accepted my Terms. It is time for Hamas to accept as well.
I have warned Hamas about the consequences of not accepting. This is my last warning, there will not be another one!
A senior Palestinian official familiar with the ceasefire efforts told the BBC that under the US proposal, all of the hostages would be released within the first 48 hours of a 60-day truce in exchange for Palestinian prisoners in Israeli jails, including those serving life sentences and other lengthy terms, and detainees from Gaza.
According to the official, the proposal also includes a personal guarantee from the US president that both sides would hold good-faith negotiations on an end to the war.
During the first two weeks of the truce, the negotiations would cover key issues, including Hamas's disarmament, the formation of an independent governing body or administrative committee in Gaza, and arrangements for Israeli troop withdrawals, the official said. Humanitarian aid would also flow openly into Gaza.
The framework appeared to be aimed at creating a pathway to a broader settlement while addressing immediate humanitarian and security concerns.
On Monday, Israeli Foreign Minister Gideon Saar told a news conference: President Trump said it clearly yesterday, Israel said yes to his proposal. We are ready to accept a full deal that would end the war based on the cabinet's decision.
Two things must happen. One, the return of our hostages… [Two], Hamas must lay down its arms.
The Israeli military has been intensifying its air and ground attacks on Gaza City, and on Monday a fourth high-rise building was destroyed in an air strike in as many days.
The international community continues to plead for quick humanitarian solutions as the humanitarian situation collapses in Gaza.
