Israeli helicopters struck Rafah on Thursday, residents said, with militants reporting street battles in the southern Gazan city as US President Joe Biden called Hamas the "biggest hang-up" to another truce.
Tensions were also soaring on Israel's northern border, with more attacks by Hamas ally Hezbollah targeting military positions and a civilian reported killed in an Israeli strike in Lebanon.
Israeli ground forces have operated in Rafah since early May, despite widespread alarm over the fate of Palestinian civilians there and an International Court of Justice ruling later that month.
Western areas of Rafah came under heavy fire on Thursday, residents said.
"There was very intense fire from warplanes, Apaches (helicopters) and quadcopters, in addition to Israeli artillery and military battleships, all of which were striking the area west of Rafah," one told AFP.
Hamas said its fighters were battling Israeli troops on the streets of the city near the besieged Gaza Strip's border with Egypt.
In Italy at a G7 summit, Biden called Hamas "the biggest hang-up so far" to a deal on a Gaza truce and hostage release.
"I've laid out an approach that has been endorsed by the UN Security Council, by the G7, by the Israelis, and the biggest hang-up so far is Hamas refusing to sign on even though they have submitted something similar," he told reporters.
"Whether or not that comes to fruition remains to be seen," he said.
The war began after Hamas's unprecedented October 7 attack on southern Israel, which resulted in the deaths of 1,194 people, mostly civilians, according to an AFP tally based on Israeli official figures.
The militants also seized 251 hostages. Of these, 116 remain in Gaza, although the army says 41 are dead.
Israel's retaliatory military offensive has left at least 37,232 people dead in Gaza, also mostly civilians, according to the Hamas-ruled territory's health ministry.
- Ceasefire push -
Efforts to reach a truce stalled when Israel began ground operations in Rafah, but Biden in late May launched a new effort to secure a deal.
On Monday, the UN Security Council adopted a US-drafted resolution supporting the plan, and on Thursday, German Chancellor Olaf Scholz said G7 leaders "call on Hamas in particular to give the necessary consent".
Some Gazans have also called on Hamas to do more to secure an agreement.
"What are you waiting for? The war must end at any cost," said a man called Abu Shaker.
Biden's roadmap for the first truce since a week-long pause in November includes a six-week ceasefire, a hostage-prisoner exchange and Gaza reconstruction.
Hamas responded to mediators Qatar and Egypt late Tuesday. US Secretary of State Antony Blinken, who was in the region this week, has said some of its proposed amendments "are workable and some are not".
Senior Hamas official Osama Hamdan said the group sought "a permanent ceasefire and complete withdrawal" of Israeli troops from Gaza, demands repeatedly rejected by Israel.
Blinken has said Israel is behind the plan, but Israeli Prime Minister Benjamin Netanyahu, whose far-right government allies strongly oppose the deal, has not publicly endorsed it.
In Jerusalem, a student-led protest near Israel's parliament urged the government to secure a hostage release deal.
"Ceasefire now," said one banner as demonstrators marched with portraits of some of the hostages.
- 'No Eid spirit' -
The war has caused widespread destruction in Gaza, with hospitals out of service and the UN warning of famine.
A UN investigation concluded Wednesday that Israel had committed crimes against humanity during the war, while Israeli and Palestinian armed groups had both committed war crimes.
The World Health Organization said more than 8,000 children aged under five had been treated for acute malnutrition in Gaza.
As Muslims worldwide prepare to celebrate Eid al-Adha beginning Sunday, displaced Gazan Umm Thaer Naseer said "we do not have anything to prepare" for the occasion.
"The children ask their father to buy clothes" for the holiday, she said in Beit Lahia in northern Gaza, adding that prices of anything from basic commodities to toys have soared.
"Where will their father buy them from? He has been unemployed for eight months and moves from one tent to another... Their father can barely feed himself."
Another displaced Gazan, Fadi Naseer, told AFP that "in normal times" homes and streets are decorated for the festival, but "today we don't even have a house anymore, and there is nothing to decorate".
"There is no Eid spirit," he added.
- Regional 'danger' -
Fallout from the Gaza war is regularly felt on the Israeli-Lebanon frontier, where deadly cross-border exchanges have escalated.
Hezbollah on both Wednesday and Thursday said it had attacked military targets in Israel with barrages of rockets and drones in retaliation for an Israeli strike that killed one of its commanders.
The Israeli military said most launches had been intercepted while others ignited fires. A government spokesman said: "Israel will respond with force to all aggressions by Hezbollah".
Later, Lebanon's National News Agency reported that Israeli "warplanes launched a raid targeting a house" in the country's south, killing one civilian and injuring seven others.
Iraqi Foreign Minister Fuad Hussein said the potential "expansion of the war is a danger, not only for Lebanon but for the entire region".
France has been making diplomatic efforts to contain the situation on the border since January, with President Emmanuel Macron saying Thursday that his country, the United States and Israel would work together to ease tensions in the area.
"We will do the same with the Lebanese authorities," he added, speaking at the G7 summit.
In the occupied West Bank, where violence has also soared during the war, Palestinian officials said an Israeli military raid killed three people in the northern town of Qabatiyah.
The army said its latest "counterterrorism operation" targeted "two senior wanted suspects".