Most analysts expect at least the first phase of the contest, among Conservative lawmakers, to be completed before Parliament breaks for its summer vacation, toward the end of July.

The statement from Mr. Brady sets the stage for a turbulent start to next month when Mrs. May has promised to hold a vote on legislation enacting her plan to leave the European Union, a proposal that has already been rejected three times by lawmakers.

The plan would keep Britain closely tied to the European Union at least until the end of 2020, and then remove it from the bloc’s main economic structures.

A critical vote is scheduled for the week of June 3, when President Trump is expected in Britain on a state visit and there will be events to mark the 75th anniversary of the D-Day landings that preceded the Allied victory in World War II.

Mrs. May is used to drama after a tumultuous period in Downing Street that has been marked by her sometimes shambolic efforts to extract Britain from the European Union, the one task she really set herself. Though her critics see her as stubborn and inflexible, her determination to plow on through adversity, her ability to absorb political pain and her refusal to stand aside have won her some admiration.

But she now appears to be nearing the end of the road. Twice, Mrs. May has had to ask the European Union to delay Brexit, and now her Conservatives face fierce competition from a new Brexit party started by Nigel Farage, which argues that Britain should leave the bloc soon, without any agreement if necessary.

Earlier this year Mrs. May offered to stand aside if Parliament voted to agree to her Brexit plan, and to allow someone else to take over the next phase of the negotiations with Brussels, which would cover the details of any new trade agreement.