Mrs. May formally resigned as leader of the Conservative Party on Friday, opening the way for a succession race. Candidates have until Monday to announce their intention to run. Conservative lawmakers will then narrow the field to two, and the party’s membership will vote.

Hanging over the contest is the deadline for Britain to exit the European Union. Boris Johnson, the former foreign secretary, is considered the top contender to become party leader, and the next prime minister. On Friday, Britain’s high court dismissed a case seeking to prosecute him over accusations that he had lied to the public during the 2016 referendum on Brexit.

Mr. Gove, the environment minister, campaigned for Britain to leave the bloc and has largely backed Mrs. May’s strategy. He was a vocal supporter of Mrs. May, and had urged others in the party to support several iterations of her deal. But through several votes in Parliament, lawmakers repeatedly rejected the plan, leading Mrs. May to step down.

In 2016, when he was a contender for the post, Mr. Gove cast himself as a reluctant candidate who was aware of his own limitations. He said he “did almost everything not to be a candidate for the leadership of this party.”

“Whatever charisma is I don’t have it; whatever glamour may be I don’t think anyone could ever associate me with it,” he said.

Mr. Gove has been a minister for justice, education and youth in the past, and has been in charge of issues like fighting drug abuse in British prisons.

By Saturday morning, Mr. Gove’s name was the top trending term on Twitter in Britain. Users posted memes and photoshopped images of Mr. Gove taking cocaine, as well as stories from the Saturday editions of the country’s main newspapers.