Egypt's state-affiliated Al-Qahera News TV channel also confirmed the arrival of the Hamas delegation in Cairo.
"The results today will be different," one Egyptian security source told Reuters.
"We have reached an agreement over many points, and a few points remain."
For the sake of the people of Gaza, the hostages & their families, and the region & the wider world – I strongly encourage the government of Israel & Hamas leadership to reach an agreement in their negotiations.— António Guterres (@antonioguterres) Without that, I fear the war will worsen exponentially.May 3, 2024
A Palestinian official with knowledge of the mediation efforts sounded cautious optimism.
"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," the official, who asked not to be named, told Reuters.
The Hamas delegation arrived from the Palestinian Islamist movement's headquarters in Qatar, which, along with Egypt, has tried to mediate a follow-up to a brief November ceasefire.
Washington - which, like other Western powers and Israel, brands Hamas a terrorist group - has urged it to enter a deal.
The talks have stumbled, however, over Hamas' long-standing demand for a commitment to end the almost seven-month-old offensive by Israel, which insists that after any truce it would resume operations designed to disarm and dismantle the faction.
"Israel will under no circumstances agree to ending the war as part of a deal to free our hostages," an Israeli official said on Saturday, signalling this core position was unchanged.
Signalling a possible breakthrough, 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.
CIA director William Burns, in Cairo for Gaza peace talks, has signalled there could be progress. (AP PHOTO)
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 could 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 rounds of truce talks and Washington has signalled there could be progress this time.
The CIA declined to comment on Burns' itinerary.
Egypt made a renewed push to revive negotiations in April, alarmed by the prospect of an Israeli assault against Hamas in Rafah in southern Gaza, where more than one million Palestinians have taken shelter near the border with Egypt's Sinai Peninsula.
A major Israeli operation in Rafah could deal a huge blow to fragile humanitarian operations in Gaza and put many more lives at risk, according to United Nations officials.
Israel says it will not be deterred from taking Rafah eventually, and is working on a civilian evacuation plan.
Saturday's Cairo talks come as Qatar reviews its role as mediator, according to an official familiar with Doha's thinking.
Qatar could cease hosting the Hamas political office, said the official.
The war began after Hamas staged a cross-border raid on October 7 in which 1200 people in southern Israel were killed and 252 hostages taken, according to Israeli tallies.
More than 34,600 Palestinians have been killed, 32 of them in the past 24 hours, and more than 77,000 have been wounded by Israeli fire during a campaign that has laid waste to the coastal enclave, according to Gaza's health ministry.
