In a meeting with legislators from her Conservative party on Wednesday, May said she would quit if her twice-defeated divorce deal passes at a third vote, which the government hopes to hold later this week.
"We need to get the deal through and deliver Brexit," the prime minister said, according to a statement released by her office. "I am prepared to leave this job earlier than I intended in order to do what is right for our country and our party."
It is the first time the embattled leader has acknowledged she is prepared to resign in order to secure the votes for her Brexit deal to pass. The prime minister did not set a date for her departure, but a leadership contest is likely to take place after May 22, her office said.
"I know there is a desire for a new approach - and new leadership - in the second phase of the Brexit negotiations, and I won't stand in the way of that," May said.
Later on Wednesday, MPs voted on eight possible alternative approaches to the prime minister's Brexit deal, but none of the options won majority support.
A motion which called for a confirmatory public vote to approve a Brexit deal was backed by 268 MPs, while 295 legislators voted against it.
A plan to stay in a customs union with the EU after Brexit received 264 votes in favour, while 272 MPs voted against it.
The opposition Labour party's plan, calling for a customs union and close alignment with the EU's single market was defeated by 307-237.
The legislator behind Wednesday's indicative votes said he was disappointed parliamentarians had failed to find a majority for the alternative approaches to Brexit.
"It is of course a very great disappointment that the House (of Commons) has not chosen to find a majority for any proposition," Oliver Letwin said, adding it was what he had expected to happen.
"If on Monday the House is able to reach a majority view I think that would be in the interests of our constituents."
Signs of support for May's deal
Britain was due to leave the EU on March 29 but has been granted a delay until April 12, after Parliament overwhelmingly rejected May's agreement on two occasions. If Parliament votes to approve her deal before April 12, the country will be granted an extension until May 22 to leave the European bloc.
Several MPs who previously voted against the deal have suggested they could now support it, if May produces a timetable for her resignation, allowing a new leader to take over negotiations over Britain's future relationship with the EU.
Speaking after May's announcement, Jacob Rees-Mogg, a leading Brexiteer in the Conservative party, said he would back the prime minister's deal if the Northern Irish Democratic Unionist Party (DUP) that props up the government abstains on the issue.
Rees-Mogg had previously said he would vote for the deal if the DUP also joins him in voting for May's divorce deal with the EU.
"If the DUP abstained I would feel entitled to back it," he told reporters. "If the DUP was still against it I would not feel able to back it."
However, the Northern Irish party later said it would not support May's deal and rejected media reports that it might abstain.
"We will not be supporting the government if they table a fresh meaningful vote," the party said in a statement.
Reporting from London, Al Jazeera's Paul Brennan said the prime minister's pledge to step down appeared to have won over some legislators who had previously rejected her agreement.
"It seems that the sands are shifting," he said. "There are several Brexiteer Tory MPs who now appear more willing to support the prime minister's deal than they previously were."
"That said, there is a hard core of the European Research Group (ERG), these really purist Brexiteers, who are still not budging. There is an estimate of around 30 of those MPs who are standing firm that they won't vote for the prime minister's deal under any circumstances."
The prime minister did not set a date for her departure, but a leadership contest is likely to take place after May 22, her office said in a statement to Conservative Party legislators.
The timetable would be set by the party and May would remain as prime minister until her successor was elected, it said, according to the Reuters news agency.
"This is about providing new leadership for phase 2 (of the negotiations to leave the EU), but we have to get through phase 1 first and leave," the statement said.
Tom Hamilton, associate director at MHP Communications, told Al Jazeera that the shape of Brexit could change significantly under a new prime minister.
"If Tory MPs are prepared to back the deal, I think it tells us a couple of things. It tells us that Tory MPs never had that principled of an objection to the deal in the first place, if they are prepared to back it in return for getting a new prime minister. Secondly, it tells us that they think that a new prime minister might give them what they want compared to what Theresa May might have."
'Not about public interest'
Britain's main opposition Labour leader Jeremy Corbyn said the prime minister's offer to step down showed her Brexit talks were "about party management, not ... the public interest".
"May's pledge to Tory MPs to stand down if they vote for her deal shows once and for all that, her chaotic Brexit negotiations have been about party management, not principles or the public interest," he said on Twitter.
"A change of government can't be a Tory stitch-up, the people must decide."
Scottish First Minister Nicola Sturgeon said the deal was "so bad that the prime minister has to promise to resign to get it through", adding that May's promise would "make an already bad project even worse".
Source: Al jazeera