It will be the fourth time that the House of Commons votes on May's Brexit deal, after rejecting it in each prior vote.
Steve Barclay acknowledged that the vote on the Withdrawal Agreement Bill would mark the end of the road for the deal struck between Mrs May and the European Union previous year unless the government could muster a majority.
The government confirmed it will bring forward the Withdrawal Agreement Bill, after Mrs May and Labour Party leader Jeremy Corbyn held fresh talks yesterday evening.
"No leader can bind his or her successor so the deal would likely be at best temporary, at worst illusory", said the letter, whose signatories included Gavin Williamson, who was sacked as defence minister this month, and former foreign minister Boris Johnson.
"Without a compromise agreement, we would be voting on the same botched Brexit deal".
Speaking after the meeting between Corbyn and May, a Labour party spokesperson said: "The Labour leader set out the shadow cabinet's concerns about the Prime Minister's ability to deliver on any compromise agreement".
"Tomorrow talks will continue at an official level as we seek the stable majority in Parliament that will ensure the safe passage of the Withdrawal Agreement Bill and the UK's swift exit from the European Union".
Setting a date for the next parliament vote without an agreement with Labour is a gamble - if she loses, May can not hold another in the current parliamentary session.
"The Prime Minister's team agreed to bring back documentation and further proposals tomorrow".
But after more than four weeks of talks, the two party leaders appear no closer to agreeing a common position, with Labour saying May had not shifted her position and warning a future Conservative leader could rip up any deal they struck.
Ms. May will bring a Withdrawal Agreement Bill, which implements the departure terms, in the week beginning June 3 - the same date that U.S. President Donald Trump begins a state visit to Britain at the invitation of Queen Elizabeth.
"We have made clear we can not support the Brexit legislation without an agreement".
Barclay said that would leave parliament with the choice of revoking the decision to leave the European Union or exiting without a deal, the default position if no divorce agreement can be reached. May's deal will keep the United Kingdom trapped within the EU's orbit for years and that it could ultimately pull the British province of Northern Ireland towards the bloc.
"If the prime minister brings the withdrawal bill to the Commons for a vote the question will be "what has changed"?" he said.
"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.
Truthfully I can not find a minister who thinks any of this will get Brexit across the line.
"She has at the same time said she would step aside once she has completed phase one, but the conversation at Cabinet wasn't about that".
Shailesh Vara, the junior Northern Ireland minister, became the first member of May's government to resign over the deal on Thursday.