The latest round of Gaza ceasefire talks have ended in Doha without a breakthrough, but a new date next week has been set for further negotiations to attempt to end the 10-month-old war.

A White House statement signed by the co-mediators Qatar and Egypt described a fresh proposal that built “on areas of agreement” and bridged remaining gaps in a manner that allowed for “a swift implementation of the deal”.

In another statement late on Friday, Joe Biden said the “bridging proposal” offered the basis for a final agreement on a ceasefire and hostage release deal, adding: “With the comprehensive ceasefire and hostage release deal now in sight, no one in the region should take actions to undermine this process.”

Though the two statements struck an optimistic tone, dozens of rounds of indirect talks between Hamas and Israel have failed to achieve a deal since a short-lived truce collapsed in December.

The hopeful language may also be aimed at further stalling Iranian retaliation against Israel after the assassination of the Hamas political leader, Ismail Haniyeh, in Tehran last month.

There are widespread concerns that an attack on Israel by Iran could spark an intense regional conflict.

The US president, Joe Biden, said: “We’re not there yet,” but added that a Gaza ceasefire deal was “closer than three days ago”. However, a Hamas spokesperson accused Washington of trying to create a “false atmosphere” without any genuine intention of stopping the war.

The new push for an end to conflict came as the Palestinian death toll in Gaza climbed past 40,000, according to local health authorities.

Diplomatic pressure has mounted on Israel to make concessions when talks resume in Cairo next week.

Speaking in Tel Aviv on Friday after a meeting with his Israeli counterpart, Israel Katz, David Lammy, the UK foreign secretary, said Israeli officials told him they were hoping they were on the verge of reaching a deal.

“The time for a deal for those hostages to be returned, for aid to get in in the quantities that are necessary in Gaza and for the fighting to stop is now,” Lammy said.

Speaking alongside him, the French foreign minister, Stéphane Séjourné, said this was a key moment “because it could lead to peace or war”.

Katz said in a statement that Israel expected its allies not just to help it defend itself against an Iranian attack, but to join in attacking Iran afterwards.

The death of Haniyeh came hours after an Israeli strike killed Fuad Shukr, a top military commander in Lebanon’s Iran-backed Hezbollah movement, which has exchanged near-daily cross-border fire with Israeli forces.

Hezbollah has promised to avenge the killing of Shukr and, in a clear message to Israel, on Friday released a video, with Hebrew and English subtitles, that appeared to show underground tunnels where trucks were transporting long-range missiles.

Israel’s military offensive in Gaza continued on Friday. Israeli forces issued evacuation orders to people in southern and central areas previously designated as “humanitarian safe zones”, saying Hamas had used the areas to fire mortars and rockets at Israel.

Israel said warning flyers and text messages had been sent out in the eastern part of Deir al-Balah and another area north of the city of Khan Younis, where tens of thousands of people have sought shelter from fighting in other parts of Gaza.

“The advance warning to civilians is being issued in order to mitigate harm to the civilian population and to enable civilians to move away from the combat zone,” the military said in a statement.

Aya, 28, said she had “felt fear, shock and confusion” after learning of the evacuation orders, but decided to flee Deir al-Balah when she heard the sounds of bombing close by.

“Things deteriorated quickly after rumours began to spread about the approach of tanks. Everyone started screaming from panic and we ended up escaping with our children, leaving our tent and most of our belongings behind,” she said.

Any sustained assault on Deir al-Balah could force tens of thousands of people to flee again to other parts of the devastated territory.

Commenting on the new evacuation order, Unrwa, the main UN agency in Gaza, said that people remained “trapped in an endless nightmare of death and destruction on a staggering scale”.

Expectations were low for this round of ceasefire talks, which began on Thursday. Hamas, which did not participate directly in the talks, has accused Israel of adding new demands to a previous proposal that had US and international support and to which the militant Islamist organisation had agreed in principle.

Both sides have agreed in principle to the plan Biden announced on 31 May, but Hamas has proposed amendments and Israel has suggested clarifications, leading each side to accuse the other of trying to undermine chances of a deal.

Hamas has rejected many of Israel’s demands, which include a lasting military presence along the border with Egypt and a line bisecting Gaza, where Israeli forces would search Palestinians returning to their homes to root out militants. Hamas has nonetheless said it is committed to the talks.

There are also disagreements over the sequencing of any ceasefire, whether it would mark a definitive end of hostilities and how many Palestinian prisoners would be released from Israeli jails in return for about 110 remaining hostages seized by Hamas in its shock raid into Israel in October last year. Hamas-led militants killed 1,200 people, mostly civilians, in that attack.

Additional reporting by Malak A Tantesh in Gaza