Senior aides of British Prime Minister Theresa May privately believe she is "finished" and may be forced to set out a timetable for her departure if she is to win the meaningful vote on her Brexit deal.

Two senior Downing Street figures are believed to have said that May should "fall on her sword" and announce she will quit so she is able to "go with dignity".

They believe she has permanently "lost the trust of Eurosceptics" and will have to make way for a new leader after the Conservative Party conference in October.

The Prime Minister is this weekend trying to win over Eurosceptics and Democratic Unionist Party (DUP) MPs ahead of a third vote on her Brexit deal on Wednesday.

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Four Cabinet ministers including Philip Hammond, the Chancellor, yesterday held several hours of talks with the DUP in an attempt to win them over and give reassurances over the Irish backstop.

Government sources said that while extra funding for Northern Ireland was not discussed yesterday, it could be "put on the table" at a later date if they support her deal.

May and her allies hope if the 10 DUP lawmakers can be persuaded to drop their opposition, many Brexiteer Conservatives will follow, giving her Brexit deal a fighting chance of winning Parliament's backing.

Still, she faces a struggle to overturn the huge defeats for the agreement, which was rejected by 230 votes in Parliament in January and by 149 votes on Wednesday.

If her EU divorce deal is approved, May will seek a delay until June 30 to give time for Parliament to pass the legislation needed for Britain's EU exit. Britain is due to leave the EU on March 29.

She has warned lawmakers opposing the agreement that if it is rejected, Britain will need a much longer extension that could see Brexit postponed indefinitely.

Despite increasing hopes of a breakthrough next week, both ministers and senior figures in Government believe May will have no choice but to go.

The Downing Street aides made their concerns clear after May lost her second vote on her deal on Wednesday and MPs voted to take a no-deal Brexit off the table.

One of the aides said the "only way" she could win round Eurosceptics was by promising to quit by the end of the year. The other said: "She needs to go with dignity. If she sets a date for her departure she can be remembered as the woman who delivered Brexit."

Downing Street last week insisted that the Prime Minister had not considered resigning in the wake of the defeats and was "totally focused" on securing her Brexit deal.