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.