Britain has legislated to leave the European Union on March 29, 2019, after triggering Article 50 - the treaty mechanism used to exit the bloc - two years prior.
A simple majority of all lawmakers could also topple her government with a vote of no confidence, with some opposition MPs calling for such a move in parliament Monday.
This evening, May told Tory MPs she will not be leading the party into the next general election, Amber Rudd said.
If May wins, there can not be another challenge for a year.
Mrs May had earlier struck a defiant tone in a statement outside Downing Street, saying: "A change of leadership in Conservative party now would put our country's future at risk and create uncertainty when we can least afford it".
With only hours remaining for Mrs May to shore up the 158 votes she needs to remain as leader, bookmakers William Hill said they were taking no more bets on the ballot.
She still has the challenge of selling her deal and not just to MPs but to party activists who are equally aggrieved that the Brexit they wanted is not what they have got.
'If there are people in Britain who think they can get a better deal, then they are wrong, ' Rutte said.
Brexit is now not officially on the agenda of the summit on Thursday and Friday, but that's likely to change after May made a statement to Parliament on Monday to call off the vote on her deal.
The PM is then scheduled to attend a summit of European leaders on Thursday at which she is likely to press for changes to elements of her Brexit deal to try to get the support of Parliament.
May calls off doomed Brexit vote
Remain-supporting Tory MP Anna Soubry said: "She needs to change course".
If several candidates come forward, a secret vote is held among Conservative MPs to whittle down the field.
Rutte has said repeatedly that the 27 European Union member states are united in their approach to Brexit and reiterated last month, after the deal was signed by European Union ministers that it is the "best we can get". Brady chairs the Conservative party committee that oversees leadership contests. All other Conservative lawmakers can run.
Nicola Sturgeon, leader of the Scottish National Party described the Conservative Party as a "self-centred bunch".
The European Parliament's Brexit point man, Guy Verhofstadt, noted that with the canceled vote in London "we have spiraled again into a new mess", and he supported Juncker's message. The vote to determine the victor is then opened to the wider Conservative Party membership.
Husband Philip showed his support by watching from the public gallery, while Tory elder statesman Kenneth Clarke told MPs that a leadership contest within months of the March 29 date of Brexit would be "irresponsible and unhelpful".
UK Prime Minister Theresa May makes a statement in the House of Commons, in London.
They include former Foreign Secretary Boris Johnson and former Brexit ministers David Davis and Dominic Raab, who all resigned their government positions in protest of May's Brexit strategy.
Javid himself has been mentioned as a possible replacement for May.
Comey set for closed-door interview with House Republicans
Comey initially resisted the subpoena from committee chairman Bob Goodlatte (R-Va.), saying he wanted a public hearing. Comey was on the Hill after backing down in his effort to fight a subpoena to force him to testify.
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