The US is considering making a deal directly with Hamas to get the remaining four American hostages out of Gaza.

Relatives of hostages still held by Hamas have been calling for the US to break with Israel and hold talks on separate deal.

It is not clear how the deal would work and what the US would offer in exchange, but any negotiations would significantly undermine Benjamin Netanyahu’s government.

An Israeli official close to the official hostage negotiations told The Telegraph that “Washington has already began looking at” a unilateral hostage deal.

The American hostages in Gaza who are still believed to be alive are Eden Alexander, 20, Sagui Dekel Chen, 35, Omer Neutra, 22, and 74-year-old Keith Siegel.

Antony Blinken, the US Secretary of State, said nearly 90 per cent of ceasefire details had been agreed on. Tension has been rising between the White House and Mr Netanyahu’s office about a deal.

“In the coming days, we will share with Israel, and they [Qatar and Egypt] will share with Hamas, our thoughts – the three [mediators] – on exactly how to resolve the remaining outstanding questions, and then it will be time for the parties to decide yes or no, and then we’ll see,” said Mr Blinken.

Last week, Joe Biden, the US president, said the Israeli prime minister was not doing enough to reach a deal. In June, sources told US media that the president had discussed with his staff negotiating directly with Hamas if the latest round of talks failed.

Meanwhile, Mr Netanyahu has doubled down on his veto of the deal currently on the table.

Despite the deaths of six hostages in a tunnel in Gaza at the weekend, he insisted this week that Israel keeps control of the Philadelphi Corridor, a buffer zone separating Gaza from Egypt, arguing that was vital for Israel’s security because it serves as the main smuggling route for Hamas.

“You have to understand the centrality of the Philadelphi Corridor to the arming of Hamas,” Mr Netanyahu told foreign reporters on Wednesday, adding that if Israel leaves the corridor that would allow Hamas to smuggle hostages out of Gaza.

But Mr Netanyahu has been criticised for his stance on the corridor, with Yoav Gallant, the defence minister, pushing back on his claims, while others have said it has only recently become a vital issue for the prime minister.

Mr Netanuahu is “changing his mind all the time about the demands. The Philadelphi Corridor wasn’t a core demand all along,” one Israeli official said.

The White House is now said to be reconsidering the deal after the families of of hostages met Jake Sullivan, the national security adviser, on Sunday.

Channel 12 reported on Thursday that the Israel Defense Forces are preparing for the possibility that the ceasefire talks will fail, likely to mean an escalation in the north with Hezbollah.

A security source also told Channel 12 that Mr Netanyahu’s statements about the Philadelphi Corridor risked undermining Israel’s relationship with Egypt, as Cairo is vehemently opposing Israeli control of the buffer zone.

Egypt also accused Mr Netanyahu of sabotaging the ceasefire talks with his demands to control the corridor.

Arab News quoted an Egyptian government spokesman as saying that Cairo holds the Israeli government responsible for the “consequences of issuing such statements that further aggravate the situation and aim to justify the aggressive and provocative policies, which lead to further regional escalation”.