Mrs May had been hoping the talks would deliver a compromise deal in time to allow her to call off the European Parliament elections.
Mrs May can only be forcibly moved from the party leadership in a vote of no confidence by the parliamentary party, initiated by the 1922 committee of backbench MPs - and her survival in the aforementioned vote would normally guarantee her a grace period in which she could not be challenged again.
Local Conservative organizations have said they will hold a non-binding no-confidence vote in May's leadership on 15 June.
As talks between the Government and Labour over a Brexit deal resume on Tuesday, there is growing Tory pressure for Mrs May to quit.
The point is that Labour's main criticism of Theresa May's Brexit plan is that it is "blind", that it makes gives no promises or commitments about the UK's future relationship with the EU.
Farage's Brexit Party, founded earlier this year amid the ongoing stalemate over the UK's divorce deal with the European Union, is now leading the polls.
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On Monday, Treasurer of the 1922 committee Geoffrey Clifton-Brown told the BBC that the Prime Minister should announce a "road map" for her resignation after the European elections set for May 23.
Labour's Shadow Chancellor John McDonnell has said he does not trust May, after details from cross-party talks were leaked to the press, and accused her of having "blown the confidentiality" of the talks.
Her effective deputy David Lidington confirmed the elections will go ahead, but said the Government was "redoubling our efforts" to get an EU Withdrawal Agreement ratified by the start of July so the MEPs elected this month never have to take their seats. Reality Check unpacks the basics.
Will the cross-party talks get anywhere this week?
But Ms Long-Bailey said there had been no movement towards a customs union, temporary or otherwise, and would only say another referendum was "one of many options".
A senior government source says it IS possible, though, to see a way to a deal, but it is unlikely to be resolved this week - and their aim is not to create some kind of May-Corbyn Rose Garden moment (imagine!) but to set out a path to get the Withdrawal Bill to Commons with a fair wind.