The agreement follows a meeting between the Prime Minister and senior Tory MPs who are demanding a date for her departure from Downing Street, the BBC reported on Thursday.
If she loses the vote on her Brexit plan, already rejected three times, sources told the BBC she would resign.
Meanwhile, Boris Johnson has said he will run for a leader once May goes.
The Prime Minister survived a confidence vote by Conservative MPs at the end of last year and party rules mean she cannot formally be challenged again until December.
But May has come under increasing pressure to leave Downing Street this summer, amid the Brexit impasse and poor results for the Conservatives in the recent local elections in England.
Informed sources told the BBC it was “inconceivable” the Prime Minister could remain in office if MPs rejected her Brexit plans for the fourth time.
But the paragraph tucked into the short formal letter from Graham Brady to Tory MPs all but marks the end of Theresa May’s premiership and the beginning of the official hunt for the next leader of the country.
After the lines in the short note restate the prime minister’s determination to get Brexit done, it confirms in black and white that after the next big vote, in the first week of June, the prime minister will make plans with the party for choosing a successor.
Right now, the expectation is that votes will be lost (although it is not impossible, of course, that Number 10 could turn it around).
And the conversation that’s been arranged won’t just be a gentle chat about what to do next.
The chairman of the 1922 committee of Conservative MPs, Graham Brady, said he had reached an agreement over the prime minister’s future during “very frank” talks in Parliament.
He said the committee’s executive and May would meet again to discuss her future following the first debate and vote on the Withdrawal Agreement Bill in the week beginning June 3.
Graham said there was now “greater clarity” about the situation.
Asked if that meant the Prime Minister would quit immediately if MPs rejected her Brexit plans once more, he said that scenario went “beyond” what had been agreed.
MPs have rejected the prime minister’s Brexit agreement with the EU three times.
But she will have another go at gaining their support in the week beginning June 3, when the Commons votes for the first time on the EU Withdrawal Agreement Bill — legislation needed to implement her deal with the EU.