The Hamas delegation arrived from the Palestinian Islamist movement's political office in Qatar which, along with Egypt, has tried to mediate a follow-up to a brief November ceasefire amid mounting international dismay over the soaring death toll in Gaza and the plight of its 2.3 million inhabitants.
Taher Al-Nono, a Hamas official and advisor to Hamas chief Ismail Haniyeh, said meetings with Egyptian and Qatari mediators had begun on Saturday and Hamas was dealing with their proposals "with full seriousness and responsibility".
However, he reiterated the group's demand that any deal should include an Israeli pullout from Gaza and an end to the war, conditions that Israel has previously rejected.
"Any agreement to be reached must include our national demands; the complete and permanent ending of the aggression, the full and complete withdrawal of the occupation from Gaza Strip, the return of the displaced to their homes without restriction and a real prisoner swap deal, in addition to the reconstruction and ending the blockade," the Hamas official told Reuters.
An Israeli official signalled its core position on this was unchanged, saying "Israel will under no circumstances agree to ending the war as part of a deal to free our hostages".
The war began after Hamas stunned Israel with a cross-border raid on October 7 in which 1200 people were killed and 252 hostages taken, according to Israeli tallies.
Egyptian officials are alarmed by the prospect of an Israeli assault against Hamas in Rafah. (EPA PHOTO)
More than 34,600 Palestinians have been killed - 32 of them in the past 24 hours - and more than 77,000 have been wounded in Israel's military operation, according to Gaza's health ministry.
The bombardment has laid waste to much of the coastal enclave.
Before the talks began there was some optimism over a potential deal.
"Things look better this time but whether an agreement is on hand would depend on whether Israel has offered what it takes for that to happen," a Palestinian official with knowledge of the mediation efforts, who asked not to be named, told Reuters.
The United States - which brands Hamas a terrorist group - has urged it to enter a deal.
Progress has stumbled, however, over Hamas' long-standing demand for a commitment to end the offensive by Israel, which insists that after any truce it would resume operations designed to disarm and dismantle the faction.
Hamas said on Friday it would come to Cairo in a "positive spirit" after studying the latest proposal for a deal, little of which has been made public.
Israel has given a preliminary nod to terms which one source said included the return of between 20 and 33 hostages in exchange for the release of hundreds of Palestinian prisoners and a weeks-long suspension of fighting.
That would leave about 100 hostages in Gaza, some of whom Israel says have died in captivity.
The source, who asked not to be identified by name or nationality, told Reuters their return may require an additional deal with broader Israeli concessions.
"That could entail a de facto, if not formal, end to the war - unless Israel somehow recovers them through force or generates enough military pressure to make Hamas relent," the source said.
Egyptian sources said CIA director William Burns arrived in Cairo on Friday.
He has been involved in previous truce talks and the US has signalled there may be progress this time.
Egypt made a renewed push to revive negotiations late last month, alarmed by the prospect of an Israeli assault against Hamas in the southern Gaza city of Rafah, where more than a million Palestinians have taken shelter near the border with Egypt's Sinai Peninsula.
