US Secretary of State Antony Blinken says Israeli Prime Minister Benjamin Netanyahu has agreed to an American "bridging proposal" for a ceasefire deal in Gaza.
It is now up to Hamas to agree, Mr Blinken added.
The pair met in Tel Aviv for talks that Mr Netanyahu has described as "positive", with his office adding that he had reiterated his commitment to an American proposal on the release of the hostages still held by Hamas, which took into account Israel’s security needs.
Mr Blinken had earlier warned this was "maybe the last opportunity" to secure a ceasefire agreement, as the US hopes to push a deal over the finish line.
The Americans hope that could happen perhaps as soon as this time next week, but that level of optimism is not shared by the Israeli leadership or Hamas.
Each accuses the other of obstinate cynicism, and blocking a deal.
Speaking in Tel Aviv after the talks, Mr Blinken described "the fierce urgency" of progressing towards a truce and hostage release deal.
"We're never giving up", he added, saying more delays could mean more hostages could die and further obstacles could hamper any agreement.
The US secretary of state will now travel on to Egypt and then Qatar, to try and drive forward progress on a deal.
Mr Netanyahu reportedly told Mr Blinken that he planned to send a negotiating team to Cairo later this week for a new round of talks with Egyptian, Qatari and US mediators.
Meanwhile, reports from Gaza speak of a worsening humanitarian situation amid continuing Israeli military activity.
Israel said its aircraft and troops had killed “eliminated dozens of terrorists” over the past day and destroyed Hamas compounds and a tunnel network where rockets and missiles were found.
Palestinian media reported that six people had been killed in an Israeli air strike near an internet access point near the southern city of Khan Younis on Monday, and that another four were killed in a strike on a car in Gaza City, in the north.
The Israeli military launched a campaign in Gaza to destroy Hamas in response to an unprecedented attack on southern Israel on 7 October, during which about 1,200 people were killed and 251 taken hostage.
More than 40,130 people have been killed in Gaza since then, according to the territory's Hamas-run health ministry.
A deal agreed in November saw Hamas release 105 of the hostages during a week-long ceasefire in return for some 240 Palestinian prisoners in Israeli jails. Israel says 111 hostages are still being held, 39 of whom are presumed dead.
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Mr Blinken was in Israel on Monday for a series of talks with key Israeli leaders.
After one meeting - with Defence Minister Yoav Gallant - a large crowd of protesters outside could be heard chanting "SOS USA, hostage deal now" and "Blinken we trust you, bring them home". Some were holding pictures of hostages.
That sense of urgency was in Mr Blinken's messages was clear.
“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,” he said before talks with President Isaac Herzog in Tel Aviv.
“I'm here as part of an intensive diplomatic effort on President Biden's instructions to try to get this agreement to the line and ultimately over the line,” he added. “It is time for everyone to get to ‘yes’ and to not look for any excuses to say ‘no’.”
Speaking alongside him, President Herzog blamed what he called “the refusal of Hamas to move forward” with a deal.
Mr Blinken then had a three-hour meeting in Jerusalem with Mr Netanyahu, whose office said was “positive and was held in a good atmosphere”.
“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,” a brief statement added.
On Sunday, the prime minister accused Hamas of being “completely obstinate” and insisted that “pressure needs to be directed” at the group - which Israel, the US and other countries proscribe as a terrorist organisation.
A Qatar-based member of Hamas’s political bureau told the BBC on Monday that it was “still interested” in reaching a deal, although he said it would not be participating in the Cairo meetings.
“We agreed a deal [through mediators] on 2 July... and therefore we don't need a new round of negotiations or to discuss the new demands of Benjamin Netanyahu,” Basem Naim said.
“We have shown maximum flexibility and positivity and the other party has understood this as a weakness and met it with more force - he is not interested in reaching a ceasefire, only in flaring up the region... and serving his own personal political interests."
The US is holding out hope that in the coming days, it can bridge the gaps on a ceasefire deal.
However, that deadline is being imposed by Washington, rather than the warring parties. And the Israeli prime minister and the leaders of Hamas do not seem to feel that same sense of urgency.
Their statements last night were very defiant, sticking to their positions and digging in.
On Tuesday, Mr Blinken will fly from Israel to Egypt, which has been an important mediator along with Qatar and is able to pass messages directly to Hamas.
The mediators announced last Friday that they had presented “a proposal that narrows the gaps between the parties” and was consistent with the principles set out by President Joe Biden on 31 May, which would run in three phases:
• The first would include a "full and complete ceasefire" lasting six weeks, the withdrawal of Israeli forces from all populated areas of Gaza, and the exchange of some of the hostages - including women, the elderly and the sick or wounded - for Palestinian prisoners held in Israel
• The second phase would involve the release of all other living hostages and a "permanent end to hostilities"
• The third would see the start of a major reconstruction plan for Gaza and the return of dead hostages' remains
The Americans have not provided details about bridging proposal, but major differences are said to remain on issues including Israel’s continuing military presence in Gaza, the rights of displaced Palestinians to move freely from north to south and the number and identity of Palestinian prisoners who’d be released from Israeli jails in exchange for Israeli hostages.
Brett McGurk, one of the Biden administration’s key envoys in the region, has been working with the Egyptians over the past couple of weeks to address the sticking point of the Philadelphi corridor, a strip of land that runs along Gaza’s border with Egypt.
Mr Netanyahu says he insists that Israel will remain present there to stop smuggling and Hamas rearming. Hamas says it simply means continued Israeli occupation and therefore not a stop to the war.
