The leaders of the US, Egypt and Qatar have called on Israel and Hamas to resume urgent negotiations to finalise a ceasefire and hostage release deal, saying there were no excuses “from any party for further delay”.

The three countries, which have been trying to mediate a deal, said in a joint statement the talks could take place in either Doha or Cairo on 15 August, adding that it was “time to bring immediate relief both to the longsuffering people of Gaza as well as the longsuffering hostages and their families”.

The leaders said a “framework agreement is now on the table with only the details of implementation left to conclude”, and offered to present “a final bridging proposal” resolving the remaining issues, such as the length of the first stage of a ceasefire and how and where Israeli troops would withdraw.

Moments after the release of the joint statement, Israel’s prime minister, Benjamin Netanyahu, said Israeli negotiators would be there. The aim, he said, was “to finalise the details and implement the framework agreement”. Netanyahu’s administration has been accused of repeatedly sabotaging ceasefire talks.

There was no immediate comment from Hamas. The announcement of the talks came after Hamas named Yahya Sinwar – the alleged mastermind of the 7 October attack – as its new leader, sparking fears that negotiations have become even more difficult. Sinwar was appointed after the assassination of Ismail Haniyeh in Iran last month.

The UK foreign secretary, David Lammy, joined the international calls for talks to resume, saying on Friday: “There can be no more delays, the fighting must stop now … all hostages still cruelly detained by Hamas must be released. We also need to see the urgent delivery of unfettered aid into Gaza.”

The flurry of diplomacy came at the same time as an increase in Israeli bombings on the Gaza Strip on Thursday, including strikes on a street in Bureij and Nuseirat camps as well as Gaza City and the southern city of Khan Younis that killed 60 people, medics said.

At least 15 people were killed in strikes on two schools functioning as shelters for people displaced from their homes in Gaza City, health officials in the Hamas-run territory said. The Israeli army said Hamas militants were using the schools as bases.

The Israeli military also renewed evacuation orders to Palestinian residents in several districts in the east of Khan Younis, Gaza’s second-largest city, saying it would act forcefully against militants who had unleashed rockets from those areas.

The army posted the evacuation order on X, and residents of the southern Gaza city said they had received text and audio messages. People said thousands had begun to leave their homes and head west towards al-Mawasi, a humanitarian-designated area but one that is overcrowded by displaced families from around the enclave.

Although Israel has designated al-Mawasi as a humanitarian zone, it has continued to bomb targets there. Khan Younis suffered widespread destruction during air and ground operations earlier this year.

Israel’s military said on Friday the new ground operation in Khan Younis was under way and troops were searching for tunnels and other infrastructure while engaging in combat “above and below ground”. Warplanes had hit 30 Hamas targets, it said.

The territory’s health ministry said the strikes in central and southern parts of the enclave had killed at least 20 Palestinians so far on Friday. Um Raed Abu Elyan told Reuters she and her family were “running from the fire, we are running with our children from fear”.

Asked where they would go, she replied: “God knows, we are walking now. They said to go to humanitarian areas, but there is no safe place here in Gaza. It is all destroyed and damaged.”

Unsanitary conditions and the destruction of healthcare facilities in Gaza mean disease is running rampant. Polio was recently detected in the territory’s wastewater, leading the WHO to announce this week it is sending more than 1m polio vaccines to the Gaza Strip. Distribution and cold storage present a “huge logistical challenge”, the UN body said.

Israeli strikes over the last 48 hours add to the mounting Palestinian death toll in Gaza – now nearing 40,000 according to the local health ministry, which does not differentiate between civilian and militant deaths. About 1,200 Israelis were killed and another 250 taken hostage – the majority civilians – in Hamas’s 7 October attack that triggered the war.

Regional tensions have soared since Haniyeh was killed on 31 July, in an attack Tehran blames on Israel. Fears are high that a retaliatory attack on Israel by Iran or its Lebanon-based ally, Hezbollah, could spark a regional war.

On Friday, Israel launched an airstrike deep inside Lebanon, killing a senior Hamas official based there, a Lebanese security source said. The strike hit the Lebanese port city of Sidon, about 60km (nearly 40 miles) from the frontier.

Separately, a senior Lebanese official said on Monday that the small country’s food supply would last four to five months under a conflict similar to the devastating 2006 war with Israel.

Nasser Yassin, the minister overseeing contingency planning for a wider conflict, told Reuters that Lebanon would need $100m a month for food, shelter, healthcare and other needs in a worst-case scenario. “A small fraction, even 10 to 15% of that, would be huge for the government. We will need donors to step up,” Yassin said.

Nearly 100,000 Lebanese people, mainly from the south, have been displaced, as well as more than 60,000 Israelis, according to official figures in the two countries.

Yassin said: “Humanitarian funding in many places has been reduced to a minimal level of just keeping heads above water. Some organisations are even slashing funding for critical lifesaving matters.”

With the heightened tensions, Cyprus – a 25-minute flight from Beirut – made clear it was willing to act as an evacuation hub in the event of the conflict spreading, as it did during the 2006 war.

More than 10 countries had already lodged requests for evacuating Europeans and third-country nationals from the volatile region to the EU’s easternmost state, Cypriot officials said. “We are ready to assist,” said the deputy government spokesperson, Yannis Antoniou.

A senior US official said the joint statement on Gaza from US, Qatari and Egyptian leaders was not designed to influence Iran, but that any escalation would jeopardise hope of getting an Israel-Hamas deal done. The official said there was no expectation that the ceasefire agreement would be signed by next week given serious issues that include the sequencing of the exchanges between Hamas and Israel. Movement was needed on both sides of the table, the person said.

A prospective cessation of hostilities has centred around a phased deal beginning with an initial truce. Recent discussions have focused on a framework outlined by the US president, Joe Biden, in late May, which he said had been proposed by Israel.

Additional reporting by Helena Smith in Athens