THERESA May will be urged by Tory grandees today to “name the day” of her departure as No 10 signalled that next month’s Commons vote on her Brexit Bill would be a make-or-break moment for her premiership.

The Prime Minister is due to meet the executive of the Conservatives’ backbench 1922 committee after saying she would leave Downing St once the first part of the Brexit process – the withdrawal agreement – is completed.

But Bob Blackman, an executive member of the 1922, said: “We need a clear timetable from the PM as to what point she will be stepping down and[when] we will elect a new leader.”

Another member of the 1922 executive, Geoffrey Clifton-Brown, added: “It would be infinitely preferable that the PM should set out this timetable herself rather than face the prospect of a vote of no-confidence in her either by her parliamentary colleagues or by association chairs.”

However, the fact Mrs May has now set a precise timescale for introducing the Withdrawal Agreement Bill[WAB] in the first week in June means her fate will hang on that date.

“Clearly, the significance of this piece of legislation can’t and I suspect won’t be underestimated,” said a Downing St spokesman.

Stephen Barclay, the Brexit Secretary, made clear before a Lords Committee that if MPs rejected the Government’s legislation at its first Commons vote at Second Reading, then Mrs May’s deal would be “dead”.

MPs, he argued, would then be faced with two remaining options as the extended delay to October 31 nears: crashing out of the EU with no deal or scrapping Brexit altogether.

MPs have, in one form or another, already rejected Mrs May’s deal three times. But the legislation is a much more detailed and, potentially, lengthier process, which will have to secure passage not just through the Commons but also the Lords. The WAB is necessary to ratify a deal and for the UK to leave the EU.

On the campaign trail in Scotland, Nicola Sturgeon said: "It doesn't look to me as if she has any chance of getting it through because nothing has changed.

"She hasn't made any progress in the last few weeks and it seems to me to be more about preserving her own party than doing what is right for the country.”

Appearing before the Lords' EU Committee, Mr Barclay said the legislation would be published "as soon as possible" and then told peers: “If the House of Commons does not approve the WAB, then the Barnier deal is dead in that form and the House will have to then address a much more fundamental question between whether it will pursue...a no-deal option or whether it will revoke."

Later, David Jones, the former Welsh Secretary, echoed the point, saying the PM’s strategy was “really high-stakes politics”.

The Tory Brexiteer claimed the PM had staked her personal prestige on the WAB. “She's had three rebuffs now and it's very hard to see where she goes after a further rebuff if the bill, when it's rejected, can't be reintroduced.

"If that bill is rejected, then it seems to me the whole policy is dead and can't be pursued any further," he added.

Yet, Mrs May might not even make it to June. Senior ministerial sources have told The Herald that her premiership is already over and that another drubbing in next week’s European elections for the Tories – widely expected by many – could see a Cabinet delegation go to the PM and tell her her time is up.

The divisive mood within the party was underlined once again during PMQs when for a second week running a Tory MP stood up and told Mrs May to her face that she should go.

Brexiteer Peter Bone said loyal and dedicated Conservative activists in his Northamptonshire constituency had made clear to him that they had lost confidence in the PM and wanted her to resign before next week’s Euro-poll.

Mrs May said she thanked all party supporters for their campaigning work and insisted her Government had been working hard to deliver Brexit, which, if most Tory MPs had voted with it, would have meant Britain would have already left the EU