Gove said Sunday that "I can confirm that I will be putting my name forward to be prime minister of this country" when May steps down as Conservative leader on June 7.
Other candidates for the top job include former Brexit Secretary Dominic Raab, Foreign Secretary Jeremy Hunt and ex-House of Commons leader Andrea Leadsom.
The current favorite is former Foreign Secretary Boris Johnson, whom Gove backed in the 2016 Conservative leadership contest before deciding to run for the job himself.
The winner of that race was May. She announced her departure Friday, admitting defeat in her three-year quest to deliver Brexit after failing three times to get her Brexit deal though Parliament.
Her successor as party leader will be selected by Conservative lawmakers and party members, and will automatically become prime minister.
The leadership contest is dominated by candidates vowing to take Britain out of the European Union even if there is no divorce deal in place.
On Sunday, Raab joined Johnson, Leadsom and another candidate, Esther McVey, in vowing to leave on the EU-set deadline of Oct. 31, deal or no deal.
"I will not ask for an extension," Raab said.
"I would fight for a fairer deal in Brussels ... and if not I would be clear that we would leave on (World Trade Organization) terms in October," he told the BBC.
Most businesses and economists think leaving the EU without agreed terms and a transition period would cause economic turmoil and plunge Britain into recession.
But many Conservatives think embracing a no-deal Brexit may be the only way to win "leave"-supporting voters back from the newly-formed Brexit Party led by Nigel Farage.
Any attempt to take Britain out of the EU without a deal will be fiercely resisted by Parliament, and could see pro-Europeans leave the Conservative Party.
"Parliament has voted very clearly to oppose a no-deal exit," Treasury chief Philip Hammond said Sunday.
"It would be very difficult for a prime minister who adopted no deal as a policy ... to retain the confidence of the House of Commons," he told the BBC.
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