The US secretary of state hopes Hamas will accept what he has called Israel’s “extraordinarily generous” offer for a Gaza truce and hostage release deal.

Antony Blinken was speaking as a Hamas delegation discussed the new proposal with mediators from Egypt and Qatar.

A source close to the talks told the BBC they were cautiously optimistic.

The proposal includes a 40-day truce in return for the release of hostages and the prospect of displaced families being allowed back to northern Gaza.

It reportedly also involves new wording on restoring calm meant to satisfy Hamas’s demand for a permanent ceasefire.

The Israeli government is coming under growing pressure from its global allies and the families of the hostages to agree a deal.

Israel launched a military campaign to destroy Hamas in response to the group’s cross-border attack on southern Israel on 7 October, during which about 1,200 people were killed and 253 others were taken hostage.

More than 34,480 people have been killed in Gaza since then, according to the territory’s Hamas-run health ministry.

A deal agreed in November saw Hamas release 105 of the hostages in return for a week-long ceasefire and some 240 Palestinian prisoners in Israeli jails.

Mediators from Egypt, Qatar and the US have been attempting for weeks to broker a new agreement that would secure another pause in the fighting and the release of the 133 hostages who Israel says are still being held, at least 30 of whom are presumed dead.

Earlier this month, Hamas rejected an Israeli proposal for a six-week truce and the release of 40 women, children and elderly or sick hostages in exchange for hundreds of Palestinian prisoners.

Hamas said it was sticking to its demands for a permanent ceasefire that would lead to a full withdrawal of Israeli troops from Gaza and the return of displaced Palestinians to their homes.

The source close to the talks in Cairo told the BBC that the new proposal from Israel was significantly different from previous offers.

On Saturday, the Axios news website cited Israeli officials as saying the proposal included a willingness for the return of people to northern Gaza and the withdrawal of Israeli troops from the east-west corridor that divides the territory and prevents freedom of movement.

It also included a willingness to “discuss the establishment of a sustainable ceasefire as part of the implementation of the second phase of the deal”, the officials said.

Israeli officials and a diplomat meanwhile told the New York Times and Financial Times on Monday that Israel was also prepared to reduce the number of hostages released during the first phase to 33, down from 40.

Hamas has only said publicly that it is studying the new Israeli proposal, but an unnamed senior official told AFP news agency on Sunday that “the atmosphere is positive unless there are new Israeli obstacles”.

“There are no major issues in the observations and inquiries submitted by Hamas regarding the contents [of the proposal],” they added.

Mr Blinken also expressed optimism at a meeting of the World Economic Forum (WEF) in Riyadh, which was attended by several of his European and Arab counterparts.

“Hamas has before it a proposal that is extraordinarily, extraordinarily generous, on the part of Israel. And in this moment, the only thing standing between the people of Gaza and a ceasefire is Hamas,” he said.

“They have to decide, and they have to decide quickly… And I’m hopeful that they will make the right decision.”

Egyptian Foreign Minister Sameh Shoukry, whose country is a mediator in the Israel-Hamas negotiations along with Qatar, also said he was “hopeful”.

“The proposal has taken into account the positions of both sides and has tried to extract moderation,” he said. “There are factors that will have an impact on both sides’ decisions, but I hope that all will rise to the occasion.”

Sunday’s phone call between US President Joe Biden and Israeli Prime Minister Benjamin Netanyahu is said to have focused on the negotiations.

They also discussed the need to sustain a recent increase in aid reaching Gaza and continued US opposition to a full-scale offensive on the southern city of Rafah, where more than a million displaced people are sheltering.

Local medics and rescuers said at least 22 Palestinians were killed in Israeli air strikes on three homes in Rafah overnight.

“We demand the entire world calls for a lasting truce. This is enough,” a man called Abu Taha told AFP at al-Najjar hospital, as a crowd of relatives mourned over the shrouded bodies.

There was no immediate comment on the reports from the Israeli military.

Meanwhile, children in Rafah told BBC Arabic’s Gaza Today radio programme that rising temperatures were making life unbearable in the thousands of tents and makeshift shelters erected there.

“Being inside the tent does not protect me from the intense heat; it is as if I am standing directly under sun’s rays,” said Sarah Abu Amr, 11.

“There is no electricity to power fans or get cold water to ease the terrible effect of the heat, and there is no food, water, or anything at all to keep us hydrated.”

Last week, when temperatures reached 40C (104F), a five-month-old girl reportedly died in a tent due to the extreme heat, according to the UN.

Over the weekend, there were further indications from senior Israeli generals that plans were being finalised for a major operation in Rafah, where the military says Hamas’s remaining battalions and leaders are based.

But Mr Blinken – who is due to fly from Saudi Arabia to Jordan and Israel – noted that the US had “not yet seen a plan that gives us confidence that civilians can be effectively protected”.

Palestinian Authority President Mahmoud Abbas – a rival of Hamas who is based in the occupied West Bank – said on Sunday that the US was the only country capable of preventing an assault on Rafah, which he warned would cause “the biggest disaster in the history of the Palestinian people”.

Israeli Foreign Minister, Israel Katz, said on Sunday that Israel’s military would “suspend the operation” in Rafah if a hostage release deal was agreed.

But far-right Finance Minister Bezalel Smotrich warned Mr Netanyahu not to cancel the Rafah assault, saying that if he failed to destroy Hamas “the government headed by you will have no right to exist”.