The prime minister's Brexit deal will be "dead" if the withdrawal bill does not pass in the Commons in June, Brexit Secretary Stephen Barclay has said.
May has promised to quit once the Brexit divorce deal is agreed, meaning she would stay in Number 10 for at least another 11 weeks.
The plan was announced during a meeting between the UK Prime Minister and opposition Labour leader Jeremy Corbyn, according to a Downing Street spokesman.
The issue of Conservatives supporting the Brexit Party has come into even sharper focus in recent days after polls revealed that the Tory party had dropped to fifth place ahead of the Euro elections, with Farage's party leading the way.
The EU Parliament elections, which take place on May 23, could provide the ideal litmus test on where public opinion is on Brexit.
May reached out to Corbyn across the Commons despite fury for many of her hard Brexit backbenchers.
The Democratic Unionist Party - the United Kingdom government's Northern Irish "confidence-and-supply" partners - have repeatedly refused to back a Brexit deal that includes the fallback customs union solution, the 'backstop'.
"I can not see how an agreement that has been rejected three times by Parliament would not be rejected a fourth time", Mr Bone told talkRADIO's Julia Hartley-Brewer.
The legal requirement to hold a meaningful vote would remain, but the government believes that if lawmakers have voted to approve the legislation, passing a meaningful vote would become a formality and could possibly be included within the bill itself.
At one point, the Prime Minister will be replaced.
This time, it will be the Withdrawal Bill which is a whole tome of new laws that will be needed to take us out of the European Union. Lawmakers are likely to object to not having the opportunity to give it enough scrutiny.
Although Theresa May might have pleaded in cabinet that people on all sides have to move away from absolutism, and move to a mood of compromise, there's not much sign of it.
Earlier today, a Cabinet source said Mrs May stressed the need for compromise and said the Government could not give in to "absolutism".
"What is the government thinking?" he asked. Presumably the PM and ministers have noticed that mortgaging the UK's Brexit future to Jeremy Corbyn is haemorrhaging the Tory vote.
"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".
May's deal covers Britain's financial liabilities, the rights of European Union citizens, proposals for a transition period and arrangements to keep open the border with Ireland after Brexit.
HuffPost's Paul Waugh tweeted: "I may be in a minority, but my own takeaway from post-PMQs briefing by Corbyn spokesman is that Labour will *not* abstain on the Withdrawal Agreement Bill 2nd reading".
Brexit has been delayed until October 31 as MPs failed to reach an agreement on how best to exit the EU.
It will also allow ministers to make "divorce payments" to the European Union foreseen under the current deal, and give effect to the so-called backstop plan for the Irish border.
May faces a punishing week when the Commons returns from recess, including the vote on the withdrawal bill, a three-day visit from Donald Trump and the Peterborough byelection, a tightly contested seat being fought by the Tories, Labour and the Brexit party.