Blinken spoke with war cabinet minister Benny Gantz and defence minister Yoav Gallant, according to statements. In both calls he "commended" Israel Israel on the proposal and "emphasised that Hamas should take the deal without delay".

Fresh Israeli strikes were reported across the Gaza Strip overnight into Monday, as mediators urged Israel and Hamas to agree to a truce and hostage release deal outlined by US President Joe Biden.

Since Biden spoke at the White House on Friday, Prime Minister Benjamin Netanyahu has insisted Israel will pursue the war -- now nearing its ninth month -- until it has destroyed Hamas and freed the Israeli captives .

Hamas has said it "views positively" what Biden described as an Israeli proposal.

Netanyahu, a hawkish political veteran leading a fragile right-wing coalition government, is under intense domestic pressure from two sides.

Protesters backing an immediate captives release, who rallied again Saturday in Tel Aviv, want him to strike a truce deal, but his far-right allies are threatening to bring down the government if he does.

Meanwhile, fighting has continued to rock Gaza, with hospitals there reporting at least 19 killed in overnight Israeli strikes into Monday morning.

Gaza's European hospital said 10 people were killed and several wounded in an Israeli air strike on a house east of the main southern city of Khan Yunis. And six people were reported killed in a strike on a family home further north in the central Bureij refugee camp, according to Al-Aqsa Martyrs hospital.

Air strikes and shelling were also reported in Gaza City, in the territory's north, as well as in Rafah, along its southern border with Egypt.

Political pressure 

Netanyahu said Saturday that "Israel's conditions for ending the war have not changed: the destruction of Hamas's military and governing capabilities, the freeing of all captives and ensuring that Gaza no longer poses a threat to Israel".

Mediators the United States, Qatar and Egypt later said they called "on both Hamas and Israel to finalise the agreement embodying the principles outlined by President Joe Biden".

White House National Security Council spokesman John Kirby told ABC News Sunday that "we have every expectation that if Hamas agrees to the proposal, as was transmitted to them -- an Israeli proposal -- that Israel would say yes".

According to Biden, Israel's three-stage offer would begin with a six-week phase that would see Israeli forces withdraw from all populated areas of Gaza and an initial captive-prisoner exchange.

Israel and the Palestinians would then negotiate for a lasting ceasefire, with the truce to continue as long as talks are ongoing, Biden said, adding it was "time for this war to end".

Netanyahu took issue with Biden's presentation, insisting that according to the "exact outline proposed by Israel" the transition from one stage to the next was "conditional" and crafted to allow it to maintain its alleged war aims.

Defence Minister Gallant, who has criticised Netanyahu over the lack of a post-war plan for Gaza, said Sunday that Israel was "assessing a governing alternative" to Hamas.

'No milk' for children 

Heavy fighting has flared in Gaza's far-southern city of Rafah, where Israel sent tanks and troops in early May, ignoring international concerns for displaced civilians there.

The UN agency for Palestinian refugees, UNRWA, said Sunday that all 36 of its shelters in Rafah "are now empty", after at least a million people fled the city.

"The humanitarian space continues to shrink", UNRWA said, adding that about 1.7 million people were now sheltering in southern Gaza's main city of Khan Yunis and in central areas.

The Palestinian Red Crescent said it was "very difficult" to access Rafah because of the Israeli bombardment.

Israel's seizure last month of the Rafah crossing has further slowed sporadic aid deliveries for Gaza's 2.4 million people and effectively closed its main exit point on the Egyptian border.

Egyptian Al-Qahera News said a Sunday meeting in Cairo with Israeli and US officials to discuss reopening the crossing had ended, without saying whether an agreement was reached.

Quoting a senior official, Al-Qahera said Egypt reiterated its demand that "Israel withdraw from the Palestinian side of the Rafah crossing so it can resume operations".

Aid agencies and the UN have warned for months of the looming risk of famine in the besieged territory.

At a hospital in Deir al-Balah, 33-year-old Amira al-Taweel told AFP that her son, suffering from malnutrition, "needs treatment and milk, but there's none available in Gaza".

"I feed him wheat (flour) which makes him bloated," she said, as her son, Youssef, lay on a narrow bed, his frail body receiving intravenous medication.

The Palestinian media office said that at least 32 people, many of them children, have died of malnutrition in Gaza since the war began.

Aid agencies say a rise in malnutrition among children is largely a result of humanitarian aid that enters Gaza not reaching its intended destination.