The US secretary of state has said during a visit to Israel that the current round of ceasefire talks is “maybe the last opportunity” to broker a truce and a hostage and prisoner swap in the 10-month-old war in Gaza.

Antony Blinken met Israeli officials, including in a three-hour one-on-one with the prime minister, Benjamin Netanyahu, on Monday during a 24-hour trip to Tel Aviv before he travels on to Egypt.

The US diplomat’s trip – his ninth since the war broke out – is part of renewed international efforts to broker a ceasefire after the recent assassinations of a top Hezbollah commander and Hamas’s political chief, Ismail Haniyeh, in Lebanon and Iran.

After the meeting, Blinken told reporters that Netanyahu “supports” the ceasefire proposal, according to the Associated Press.

“In a very constructive meeting with Prime Minister Netanyahu today, he confirmed to me that Israel supports the bridging [ceasefire] proposal,” Blinken said. “The next important step is for Hamas to say yes.”

US officials have been accused of being too optimistic in their regular claims that negotiators were on the verge of striking a deal. But on Monday Netanyahu’s office also put out a rare public statement backing the US bridging proposal.

“The prime minister reiterated Israel’s commitment to the current American proposal on the release of our hostages, which takes into account Israel’s security needs, which he strongly insists on,” the statement read.

A spokesperson for Netanyahu confirmed that the prime minister had told Blinken that Israel had agreed to the bridging proposal, the New York Times reported.

The killings of Haniyeh and the Hezbollah commander had set the Middle East on edge, and a cessation of hostilities in Gaza has been seen as the best way to cool the regional tensions. Tehran and the powerful Lebanese militia have threatened retaliatory action.

“This is a decisive moment, probably the best, maybe the last, opportunity to get the hostages home, to get a ceasefire and to put everyone on a better path to enduring peace and security,” Blinken said before a meeting with Israel’s president, Isaac Herzog.

“It’s also time to make sure that no one takes any steps that could derail this process,” he said in a veiled reference to Iran. “And so we’re working to make sure that there is no escalation, that there are no provocations, that there are no actions that in any way move us away from getting this deal over the line, or for that matter, escalate the conflict to other places and to greater intensity.”

The latest negotiations began in Doha last week and are expected to resume in Cairo on Wednesday or Thursday, but optimism from international mediators at the close of two days of Qatar negotiations has not been matched by Israel or Hamas.

In its first official comments since the new round of talks began, Hamas said on Sunday night that the latest proposal on the table was a capitulation to Israel that “responds to Netanyahu’s conditions”, negating the possibility of future talks. Hamas is not directly participating in the latest round and is instead being briefed on developments by the mediators Qatar and Egypt.

Israel, too, has expressed an unwillingness to compromise on points such as the withdrawal of troops from the Gaza-Egypt border. Netanyahu said at the start of his weekly cabinet meeting on Sunday: “There are things we can be flexible on and there are things that we cannot be flexible on, which we will insist on.”

The plan would involve an initial six-week ceasefire during which a limited number of Israeli hostages would be freed in exchange for Palestinians held in Israeli jails, and the amount of humanitarian aid entering the strip would increase.

Unlike the week-long truce that collapsed at the end of November, this ceasefire would be indefinitely extendable while negotiators work on the next stage, which would involve another round of hostage and prisoner swaps and a drawdown of Israeli troops.

There have been reports of friction between Netanyahu and his negotiating team over the future of the Gaza-Egypt border zone, known in Israel as the Philadelphi corridor. The Israeli news website Ynet reported on Monday that the prime minister had refused to countenance giving up control of the area, although members of the security establishment, including his defence minister, Yoav Gallant, believed a compromise could be reached with the use of surveillance technology.

Netanyahu’s critics have repeatedly accused him of stalling on implementing a deal in order to appease his rightwing allies, who have threatened to collapse his government. The longtime Israeli leader sees staying in office as the best way of avoiding a conviction for corruption charges, which he denies.

An Israeli delegation travelled to Egypt on Sunday to discuss the possibility of a withdrawal mechanism from Rafah, but no progress was made on the issue and the delegation did not offer anything new, an Egyptian official told Agence France-Presse.

Hours after Blinken landed in Tel Aviv there was a small explosion in the city in which one person was killed and another injured. It was later claimed as a suicide bombing by the armed wings of Hamas and Palestinian Islamic Jihad. Hamas had stopped using the tactic in 2005.

The groups threatened to carry out more such attacks in Israel “as long as the occupation’s massacres, the displacement of civilians and the policy of assassinations continue”.

Despite Blinken’s plea that “no one takes any steps that could derail” the ceasefire talks, violence continued to rage in Gaza and on the Israel-Lebanon border on Monday. An Israeli soldier and two Hezbollah fighters were killed in cross-border clashes, the Israeli military and the Lebanese militant group confirmed.

In Gaza, three people were reported killed in an Israeli airstrike in the southern village of Abassan, and airstrikes in Khan Younis killed a baby and injured several women, medics in the Hamas-dominated territory said. The Israeli military said it had struck “45 terrorist targets” in the past 24 hours and ground forces were operating in the Khan Younis area.

About 170,000 people have been displaced in the past week after the Israeli army issued new evacuation orders across southern Gaza, including for some areas previously designated as humanitarian safe zones. According to the Office for the Coordination of Humanitarian Affairs, the UN’s humanitarian agency, just 11% of Gaza’s total area is now deemed “safe”, although Israel has also bombed the humanitarian zones on several occasions.

Almost all of the strip’s 2.3 million population have been displaced from their homes and 40,000 people have been killed amid a devastating humanitarian crisis, according to the local health authority.

About 1,200 people were killed and 250 taken hostage in Hamas’s 7 October attack on southern Israel that triggered the war.