Fresh strikes were reported in the Gazan city of Rafah 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 captives taken during the Palestinian militant group's unprecedented October 7 attack.
Hamas has said it "views positively" what Biden described as an Israeli proposal.
US Secretary of State Antony Blinken called Israeli war cabinet member Benny Gantz and Defence Minister Yoav Gallant to discuss the deal, the State Department said in a pair of statements Sunday night.
In the calls, Blinken "commended" Israel on the proposal and "emphasised that Hamas should take the deal without delay".
Netanyahu, a hawkish political veteran leading a fragile right-wing coalition government, is under intense domestic pressure from two sides.
Protesters backing an immediate hostage 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 Israel's military reporting air strikes and ground combat on Sunday.
Gaza's European hospital reported late Sunday night that three people were wounded in a strike on a neighbourhood in northern Rafah, while eyewitnesses reported multiple injuries and deaths in a strike early Monday on a home west of the city.
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 hostages 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".
The Gaza war was sparked by Hamas's October 7 attack, which resulted in the deaths of 1,189 people, mostly civilians, according to an AFP tally based on Israeli official figures.
Militants also took 252 hostages, 121 of whom remain in Gaza, including 37 the army says are dead.
Israel's retaliatory bombardments and ground offensive have killed at least 36,439 people in Gaza, mostly civilians, according to the Hamas-run territory's health ministry.
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 hostage-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 war aims.
Finance Minister Bezalel Smotrich and National Security Minister Itamar Ben Gvir, leaders of the two extreme-right parties in parliament, warned they would leave the government if it endorsed the truce proposal -- potentially costing Netanyahu's coalition its majority.
But opposition leader Yair Lapid, a centrist former premier, said the government "cannot ignore Biden's important speech" and vowed to back Netanyahu if his far-right coalition partners quit.
"I remind Netanyahu that he has our safety net for a hostage deal," Lapid said on X.
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.
Heavy fighting has flared in Gaza's far-southern Rafah city, 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.
Witnesses said Israeli Apache helicopters struck central Rafah on Sunday, also reporting clashes there and air raids and shelling in other parts of the city.
The Palestinian Red Crescent said it was "very difficult" to access Rafah because of the Israeli bombardment.
Helicopter attacks and shelling were also reported in the north and centre of the territory.
In Syria, a strike attributed to Israel killed at least 12 "pro-Iranian fighters" early Monday morning near Aleppo, the Syrian Observatory for Human Rights reported.
While it rarely comments on individual strikes in Syria, Israel has repeatedly said it will not allow its arch-enemy Iran -- which also backs Hamas -- to expand its presence there.
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 state-linked 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".
Cairo refuses to coordinate with Israel humanitarian deliveries through Rafah, but has agreed to send some aid via Israel's Kerem Shalom crossing.
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 Hamas government 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.
