Talks between Brexit Secretary Dominic Raab and EU chief negotiator Michel Barnier have failed to resolve key issues including the Irish border question.

The unscheduled talks in Brussels came amid speculation officials working on the negotiations had reached a deal.

But following the meeting Mr Barnier said that “despite intense efforts” there had been a failure to reach agreement on one of the trickiest aspects of the negotiations.

The surprise announcement of the meeting fuelled rumours a deal was set to be done ahead of a summit of EU leaders on Wednesday.

But after talks which lasted a little over an hour, it was clear that obstacles remained.

Mr Barnier said “some key issues are still open”, including the so-called backstop measure to prevent a hard border.

The UK Government said there were still “unresolved issues” relating to the backstop but it remained committed to making progress at Wednesday’s summit.

A spokesman said: “In the last few days UK and EU negotiators have made real progress in a number of key areas. However there remain a number of unresolved issues relating to the backstop.”

A senior UK source suggested Brussels had sought to promote the idea a deal had been reached before the meeting so it would “look like we are the ones being intransigent”.

But with Mrs May under increasing pressure from Tory Eurosceptics and her DUP parliamentary allies, the Government also has a reason to appear to be taking a tough line.

Former Brexit secretary David Davis has urged a Cabinet revolt against the Prime Minister’s approach to the talks.

And Tory MP Nadine Dorries suggested Mr Davis should be installed as interim leader, claiming that was the only way to secure the kind of free-trade deal Brexit demanded by Eurosceptics.

Allies of Mr Davis said he has been contacted by several Tory MPs urging him to run for the leadership.

The issue of the Northern Irish backstop – a mechanism to avoid a hard border – is at the heart of the troubles facing Mrs May as she attempts to secure a deal.

The European Union’s version of the backstop, which would see just Northern Ireland remain aligned with Brussels’ rules, has been called unacceptable by Mrs May and is loathed by the DUP.

Mrs May’s counter-proposal is for a “temporary customs arrangement” for the whole UK, but Tory Brexiteers fear this becoming an open-ended position which would prevent free trade deals with countries around the world.

Foreign Secretary Jeremy Hunt, who was hosting eight EU counterparts at his Chevening country retreat, said “there is no-one who is going to be able to negotiate the right deal for Britain better than Theresa May”, adding “she is battling for Britain”.

Health Secretary Matt Hancock sought to play down the potential for Cabinet resignations but was unable to say whether a fixed deadline for any customs arrangement would be written into a deal with Brussels.

He told the BBC’s Andrew Marr Show: “There are different ways that you can make sure that something is credibly time limited and that’s what I want to see.”

Writing in the Sunday Times, Mr Davis, who quit in July over Brexit, said: “It is time for the Cabinet to exert their collective authority.”

The newspaper said at least nine ministers want Mrs May to change course when the Cabinet meets on Tuesday.

Staunch Brexiteers Andrea Leadsom, Penny Mordaunt and Esther McVey are at the centre of resignation speculation, but the report also indicated that Scottish Secretary David Mundell and Scottish Tory leader Ruth Davidson could quit because of the sensitivity of the issue in relation to calls for Scottish independence.

Mrs May’s own position also appeared in jeopardy, with as many as 44 letters demanding a vote of no confidence reportedly submitted to the Conservative 1922 Committee – just four short of the number required to trigger a ballot.

Critics of the Prime Minister believe the threshold could be passed by Wednesday, depending on the nature of her approach to the Brexit talks.

Ms Dorries backed Mr Davis for the top job, saying: “Getting May out and him becoming an interim leader may be the only way to deliver Brexit and FTA (a free-trade agreement).”

Mr Davis is understood to be prepared to run if there is a leadership contest.

DUP leader Arlene Foster will meet Taoiseach Leo Varadkar on Monday to discuss the Brexit situation, while Sinn Fein’s leader Mary Lou McDonald will hold talks in Westminster with Mrs May and Labour leader Jeremy Corbyn.

Mrs May’s hopes of getting a Brexit deal through Parliament could depend on the actions of Labour MPs, with former minister Caroline Flint telling Sky she would back a “reasonable” deal.

But shadow foreign secretary Emily Thornberry told the BBC: “If she comes back with something that’s just a fudge she’s cooked up with Brussels… we’re not voting for something that’s essentially a bridge to nowhere.”