A government spokesperson said May will put forward a Withdrawal Agreement Bill, making Brexit law in the United Kingdom, in the week of June 3, before the summer parliamentary recess in July.
Equally, her fate could be sealed if her plans are rejected for the fourth time by MPs in June.
Both May and Labour leader Jeremy Corbyn are under strong pressure from their party members not to make concessions to their rivals.
"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".
The vote is expected to take place in the week beginning June 3, Downing Street has announced.
The letter said the Prime Minister can not bind her successor, so any agreement with Labour would be "at best temporary, at worst illusory".
No date has been set for the summer recess, but Parliament usually rises near the end of July.
But Corbyn, whose negotiating team has been holding talks with government ministers for more than four weeks to find a way to break the deadlock in parliament, raised doubts over whether Labour could back the Withdrawal Bill. May has said she will step down once the first phase of Brexit is complete.
May, who secured the Conservative Party leadership and the premiership in the chaos that followed Britain's 2016 vote to leave the European Union, is under pressure from some of her own lawmakers to set a date for her departure.
The Prime Minister has been engaged in cross-party talks with Jeremy Corbyn since she asked for the latest deadline delay until October 31.
But Mr Bone said unless there were substantial changes to the Irish backstop within the Withdrawal Agreement, it would not be backed by MPs.
After warning the government has yet to put forward "the significant shift" in position the party needed to support a deal, McDonnell said: "What we are all concerned about is the letter from Boris Johnson which said one he wouldn't accept a customs union, one of our key elements, and secondly that if he was leader, and he may well be, that he would overturn any deal that we'd agreed".
"More fundamentally, you would have lost the loyal middle of the Conservative Party, split our party and with likely nothing positive to show for it".
Referring to the prospect of a new Tory leader tearing up any agreement, he added: "Our big problem now is, if we are going to march our troops in Parliament to the top of the hill to vote for a deal and then that's overturned within weeks, I think that would be a cataclysmic act of bad faith".
British PM Theresa May will have to get a Brexit deal through parliament before the summer break.
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