Theresa May will put her key Brexit Bill before MPs at the beginning of June, No 10 has announced, as she set a new deadline of mid-July to get a deal through Westminster.

The move, which appears to signal the Prime Minister's final throw of the political dice on Brexit, came after she had a "useful and constructive" meeting with Jeremy Corbyn to take stock of the cross-party talks.

A Downing St spokesman said it had taken place to "make clear our determination to bring the talks to a conclusion and deliver on the the referendum result to leave the EU".

Read more: Blunder as Highland voters sent Labour election leaflets in Welsh - not Gaelic

He went on: "We will, therefore, be bringing forward the Withdrawal Agreement Bill[WAB] in the week beginning June 3." This is the week of the state visit by US President Donald Trump and when MPs return following the Whitsun break.

"It is imperative we do so then, if the UK is to leave the EU before the summer parliamentary recess." This usually takes place in the second or third week of July.

The spokesman added: "Tomorrow talks will continue at official level as we seek the stable majority in Parliament that will ensure the safe passage of the WAB and the UK's swift exit from the EU."

Read more: Sturgeon admits to 'imposter syndrome' in FM role

The mid-July deadline for a deal could also prove to be one for the end of Mrs May's premiership although senior Conservative insiders believe this could come sooner if next week’s European elections are as bad as predicted for the party.

Ministerial sources have suggested in such circumstances a delegation of Cabinet ministers could swiftly call time on their leader and demand she goes.

After a two-hour weekly Cabinet session discussed the cross-party talks, senior ministers agreed to continue with them but with a clear view that "we need to get a move on".

The PM’s spokesman explained: "Ministers involved in the negotiations set out details of the compromises which the Government was prepared to consider in order to secure an agreement, which would allow the UK to leave the EU with a deal as soon as possible.

"Cabinet agreed to continue discussions with Labour to see what was possible.”

Read more: Curtice says SNP on track for 'all-time record' in European election

Asked if getting the deal through by the start of the summer recess would also make that Mrs May's exit date, the spokesman replied: "What she wants to do is get a deal through by the summer recess.

"She has at the same time said she would step aside once she has completed phase one. But the conversation at Cabinet wasn't about that."

A Cabinet source said Mrs May stressed to her colleagues the need for compromise and that the Government could not give in to "absolutism".

Earlier, No 10 attempted to calm Tory fears about the prospect of a damaging split in the party over a customs union; Labour's key demand in the talks.

Mrs May was warned by senior Conservatives that she risked losing the "loyal middle" of the party if she gave ground on the issue.

But Downing St sources insisted the Government would not sign up to a "permanent" customs union and any compromise position would only be an "interim" measure.

Some 13 former ministers, together with the chairman of the backbench 1922 Committee, Sir Graham Brady, wrote to the PM urging her not to concede Labour's key demand.

The signatories included the sacked Defence Secretary Gavin Williamson as well as potential leadership contenders Boris Johnson and Dominic Raab.

The letter, organised by former minister Greg Hands, said Mrs May could not bind her successor to a deal so any agreement with Labour was likely to be "at best temporary, at worst illusory".

It added: "We believe a customs union-based deal with Labour will very likely lose the support of Conservative MPs, like us, who backed the Withdrawal Agreement in March (in many cases very reluctantly), and you be unlikely to gain as many Labour MPs to compensate."

On Thursday, the PM is set to meet members of the 1922 executive in a bid to get her to name a date for her departure; they are likely to be disappointed.

In other developments:

*the latest poll by BMG shows Nigel Farage’s Brexit Party, compared to 2014, are on 26 points [+26], Labour on 22 [-3], the Liberal Democrats on 19 [+12], the Tories on 12 [-12], the Greens on 10 [+2] and Ukip on 3 [-24].

*a Kantar poll gave Labour a nine-point lead, [34 to 25], which would put Jeremy Corbyn in Downing St at a general election.

*Labour’s John McDonnell claimed of the cross-party talks: "We are not near what we want," and warned Labour's "big problem" was "if we are going to march our troops in Parliament to top of hill to vote for deal and then that's overturned within weeks, that would be cataclysmic act of bad faith".

*Jeremy Hunt, the Foreign Secretary, warned the Tories and Labour would be "crucified' by voters if they failed to resolve Brexit and respected the referendum result.

*Sir Michael Fallon, the former Defence Secretary, said: "If they are going to include permanent membership of a customs union then, frankly, we would be better off staying in the EU because at least then we would have a voice in the trade arrangements being negotiated."

*Olly Robbins, Britain’s chief Brexit negotiator, was in Brussels to talk about the possibility of changing the Political Declaration on future relationship should a cross-party deal be agreed but the EU made clear it remained on a "Brexit break" until something changed.

*Former Tory minister Crispin claimed the Conservatives were "almost certainly going to have to go into some kind of electoral arrangement with the Brexit Party, otherwise Brexit doesn’t happen," but his colleague, Sam Gyimah, another former minister, insisted such a suggestion "should send a chill down the spine of anyone who believes our party stands for sensible government in the national interest".