"Hamas informed the mediators of its formal written response, which includes a positive response to (US envoy Steve) Witkoff, but with emphasis on guaranteeing a permanent ceasefire and full Israeli withdrawal," said the Hamas source with knowledge of the ongoing negotiations.
The Palestinian group said it would release 10 living hostages and hand over the bodies of 18 dead in exchange for Palestinian prisoners and detainees held in Israeli prisons.
The Witkoff proposal includes a truce of at least 60 days.
The White House had said the latest proposal for a deal was approved in advance by Israel.
Hamas said "as part of this agreement, 10 living prisoners of the occupation held by the resistance will be released, in addition to the return of 18 bodies, in exchange for an agreed-upon number of Palestinian prisoners".
A breakthrough in negotiations has been elusive ever since a previous ceasefire fell apart on March 18 with the resumption of Israeli operations.
But US President Donald Trump had said Friday that the parties were "very close to an agreement".
Two sources close to the negotiations had said Witkoff's proposal involved a 60-day truce, potentially extendable to 70 days.
It would see the release of five living hostages and nine bodies in exchange for a number of Palestinian prisoners during the first week, followed by a second exchange the following week, the sources said.
Israel has come under increasing international criticism over the dire humanitarian situation in the Palestinian territory, where the United Nations recently warned the entire population was at risk of famine.
This week a spokesman for the UN humanitarian agency called the territory "the hungriest place on Earth".
Aid is only trickling into Gaza after the partial lifting by Israel of a more than two-month blockade, and the UN has recently reported looting of its trucks and warehouses.
The World Food Programme has called on Israel "to get far greater volumes of food assistance into Gaza faster", saying desperation was "contributing to rising insecurity".
The health ministry in Hamas-run Gaza said on Saturday that at least 4,117 people have been killed in the territory since Israel resumed its offensive on March 18, taking the war's overall toll to 54,381, mostly civilians.
The proposals would see a 60-day truce and the exchange of 28 of the 58 hostages still held in Gaza for more than 1,200 Palestinian prisoners and detainees, along with the entry of humanitarian aid into the enclave.
A Palestinian official familiar with the talks told Reuters that among amendments Hamas is seeking is the release of the hostages in three phases over the 60-day truce and more aid distribution in different areas. Hamas also wants gurantees the deal will lead to a permanent ceasefire, the official said.
There was no immediate response from Israeli Prime Minister Benjamin Netanyahu's office to the Hamas statement.
Yesterday, aid groups said dozens of World Food Programme trucks carrying flour to Gaza bakeries had been hijacked by armed groups and subsequently looted by people desperate for food after weeks of mounting hunger.
"After nearly 80 days of a total blockade, communities are starving and they are no longer willing to watch food pass them by," the WFP said in a statement.
Hamas response unacceptable, says US envoy
The US envoy to the Middle East on Friday criticized Hamas over its response to a US-proposed ceasefire deal. "It is totally unacceptable and only takes us backward," Steve Witkoff wrote on X. "Hamas should accept the framework proposal we put forward as the basis for proximity talks, which we can begin immediately this coming week."That is the only way we can close a 60-day ceasefire deal in the coming days... and negotiations in good-faith to try to reach a permanent ceasefire," he added.
