Hopes rose on Monday for a long-sought truce and hostage release deal after almost seven months of Israel's war on Gaza.
Washington's top diplomat said he was "hopeful" Hamas would accept the offer, which his British counterpart said could see the release of thousands of Palestinian prisoners.
Negotiators from Hamas were due in Egypt, a mediator in the process along with the United States and Qatar.
For months they have been trying to broker a new agreement between the combatants, the first since a one-week truce in November saw 80 Israeli hostages exchanged for 240 Palestinians held by Israel.
Diplomacy in the past few days appeared to suggest a new push to halt the fighting.
The war has brought Gaza to the brink of famine, United Nations and humanitarian aid groups say, while reducing much of the territory to rubble and raising fears of a wider regional conflict.
US Secretary of State Antony Blinken told a World Economic Forum (WEF) special meeting in Riyadh that the proposal before Hamas is "extraordinarily, extraordinarily generous on the part of Israel".
He urged the group to "decide quickly", saying: "I'm hopeful that they will make the right decision."
Blinken is on his seventh visit to the region since the Gaza war began on 7 October.
Talks "are taking place in Cairo today", said Al-Qahera News, which is linked to Egyptian intelligence services.
A senior Hamas official said on Sunday the group had no "major issues" with the latest truce plan.
Speaking at the WEF meeting, Egypt's Foreign Minister Sameh Shoukry said "the proposal has taken into account the positions of both sides".
While there was no "final decision" yet, Shoukry said: "We are hopeful… I hope that all will rise to the occasion."
British Foreign Secretary David Cameron told the same gathering that Hamas has been offered a "sustained 40 days' ceasefire, the release of potentially thousands of Palestinian prisoners, in return for the release of these hostages".
Saudi Foreign Minister Faisal bin Farhan meanwhile said a new deal would "be very, very positive… But it's absolutely necessary that any ceasefire be permanent, not temporary".
Israel's war on Gaza has killed at least 34,488 people, according to the Palestinian enclave's health ministry.
The tally includes at least 34 deaths in the past 24 hours, the ministry said. That is down from a peak this month of at least 153 deaths on 9 April.
A Hamas-led 7 October attack on Israel resulted in the deaths of about 1,170 people in Israel, mostly civilians, according to an AFP tally based on Israeli official figures.
The attack saw hostages seized, 129 of whom Israel estimates to remain in Gaza, including 34 the military says are dead.
Away from the chandeliered WEF meeting hall, southern Gaza's Rafah city was pounded by airstrikes overnight, an AFP correspondent, witnesses and rescuers reported.
The majority of Gaza's 2.4 million people have sought refuge in Rafah near the border with Egypt.
Medics and the Civil Defence agency said at least 22 people were killed in the city. Witnesses told AFP at least three houses had been hit.
A crowd of grief-stricken relatives jostled over the dead, shrouded in white, at the city's Al-Najjar Hospital. Some women gently stroked the body bags holding the remains of their loved ones.
"We demand the entire world to call for a lasting truce. This is enough," a grey-haired relative who gave his name only as Abu Taha said at the hospital.
A Hamas source close to the talks has told AFP the group is keen for a deal that "guarantees a permanent ceasefire, the free return of displaced people, an acceptable deal for [a prisoner-hostage] exchange and an end to the siege" in Gaza.
In Israel, protesters have demanded that the government reach an agreement to secure the release of the hostages still held in Gaza.
Israel has pledged to attack Rafah despite mounting global concern for civilians there, but Foreign Minister Israel Katz said the government may "suspend" the invasion if an agreement is reached.
High temperatures in crowded Rafah have turned makeshift shelters made from plastic tarps into sweltering ovens. The UN has warned of diseases spreading.
"We're living in hell," said displaced Palestinian Hanan Saber, 41.
On Sunday, White House National Security Council spokesman John Kirby said on ABC News that Israel is letting more aid trucks into Gaza, in line with "commitments that President Biden asked them to meet".
After an Israeli drone strike in early April killed seven workers from a US-based charity, US President Joe Biden suggested to Israeli Prime Minister Benjamin Netanyahu, for the first time, that continued US support could be conditional on protection and aid for civilians.
Most of the World Central Kitchen (WCK) workers killed were Westerners. Israel said the strike was a mistake.
WCK had supplied meals in Gaza and had just helped open a maritime aid corridor, but suspended work after the strike.
On Sunday, WCK said it was resuming operations in Gaza.
According to the Hamas source, the latest plan proposes an Israeli withdrawal from two main roads to allow Gazans to return to the heavily impacted north.
Hamas negotiator Zaher Jabareen told AFP that "success or failure" will be determined by "the ability to reach a permanent ceasefire decision" – a condition Israel has rejected before – and to agree "clear" plans for reconstruction.
The Axios news website, citing Israeli officials, reported that Israel's latest proposal includes a willingness to discuss the "restoration of sustainable calm" after hostages are released.
It is the first time Israeli leaders have suggested they are open to discussing an end to the war, Axios said.
Biden spoke with Netanyahu by phone on Sunday about the truce negotiations and "increases in the delivery of humanitarian assistance into Gaza", the White House said.
They discussed preparations to open new crossings to northern Gaza, where conditions have been particularly dire, it added.
