While European Union elections have always been seen as a safe place for mid-term protest votes to take place by the political establishment in the United Kingdom, polling showing that the month-old Brexit Party is now beating the Conservatives in even general election polling has sent shockwaves through Westminster.
Labour and the Tories have been locked in talks for weeks to try and break the political paralysis gripping Westminster.
Asked whether the vote would be considered a "confidence" vote for the Prime Minister, a source said: "That's not the world that we are now in but clearly the significance of this piece of legislation can't and I suspect won't be underestimated".
May will bring a Withdrawal Agreement Bill, which implements the departure terms, to parliament for a vote in the week beginning June 3, Brexit Secretary Steven Barclay said.
May and opposition leader Jeremy Corbyn sat down for talks again on Tuesday ahead of further discussions between the government and opposition on Wednesday.
On Thursday, she will meet senior Conservatives who have demanded a detailed timetable for her departure. "It is imperative we do so then if the United Kingdom is to leave the European Union before the summer parliamentary recess", No 10 said.
Lawmakers have already rejected May's Brexit deal with the European Union three times amid deep divides over when, how or even if the divorce will take place.
They also said it was "imperative" that any exit plan be approved by parliament before MPs go on their summer holiday, which normally begins at the end of July.
Labour Party leader Corbyn said last week that May had made no big offer on Brexit and had not moved her "red lines". I can not imagine for one minute that Theresa May is going to want to go to the next Conservative Party conference in October and address it as leader. Labour's main demand is for a permanent customs union that can not be ripped up by a Brexiteer, such as Boris Johnson, the moment May steps down.
A compromise between the two parties was not impossible but talks could not continue "indefinitely", according to Foreign Secretary Jeremy Hunt.
The Cabinet minister insisted Britain does not need another referendum or general election.
Following another round of cross-party talks with the Labour Party on Tuesday night, Downing Street issued a statement to reveal details of yet another Brexit vote.
"Because both of us will be crucified by our base if we went into a general election having promised that we would respect the referendum result, not having respected it".
Shadow chancellor John McDonnell, speaking at a Wall Street Journal event yesterday, said it was becoming hard to see how Labour could "march our troops up the hill" when Boris Johnson and Dominic Raab were prepared to overturn any cross-party deal in what he said would be a "cataclysmic act of bad faith".
"I don't think it's impossible that there could be a deal there".
"I think 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" the entire Brexit process, Brexit Secretary Steve Barclay told the Lords EU Committee.
"It is plain that her proposals are not going to get through the House of Commons anyway, so I am afraid it is going to be her successor who decides it".