But speaking outside Downing Street, May said that she meant to contest the confidence vote "with everything I've got", adding that she was ready to "finish the job" she began after becoming party leader.
Legislators began casting their secret ballots in a wood-panelled room in Parliament after May's Conservative opponents, who have been circling for weeks, finally got the numbers they needed to force a vote of no confidence.
If she loses the confidence vote, Mrs May must step down and there will be a contest to choose a new leader.
In a statement Sir Graham said: 'The votes will be counted immediately afterwards and the result will be announced as soon as possible in the evening'. Now the PM's own party may sack her.
"The only way out of this mess is for the PM to show real leadership by calling a People's Vote, and letting the public choose between this Brexit chaos and uncertainty or stability under the best deal we already have as a member of the European Union". "Prime Minister, take responsibility, do the right thing, resign".
One cabinet minister said that the PM had tried to reassure MPs that she would not do anything to break faith with the DUP.
The ballot will be held on Wednesday evening between 6pm and 8pm local time.
Each time the politician with the lowest number of votes is removed until just two candidates remain.
After a day of high drama at Westminster, May defeated the challenge to her authority - but only after making an emotional pledge that she would not lead her party into the next general election.
"Conservatives must now answer whether they wish to draw ever closer to an election under Mrs May's leadership".
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Burnham-On-Sea's MP has today given his backing to the Prime Minister and said he will be supporting her in tonight's "no confidence" vote.
Mrs May has said she is seeking guarantees that this clause will be "temporary" but the Democratic Unionist Party, which she relies on to win key votes, wants the backstop to be ditched altogether.
May said before the vote that it was "now clear" the backstop needs to be temporary, and that she was confident she could still win approval in Parliament for her withdrawal plan with further assurances from the EU.
The announcement came after Brexiteer Tories, who had to date held back from submitting letters calling for a confidence vote, broke cover in anger at the shelving of the vote on the Brexit deal. She warned a leadership change now would delay or even imperil Britain's planned divorce from the European Union, which is scheduled for 29 March. None of that would be in the national interest.
"So here is our renewed mission, delivering the Brexit that people voted for, bringing the country back together, and building a country that truly works for everyone".
The pound fell briefly in response to news of the challenge to Mrs May's position, but later rallied.
The challenge has been brought by Conservative MPs who think Mrs May has watered down the Brexit voters were promised in the 2016 referendum.
"I will be backing Theresa May tonight".
He added that the Conservative Party's "internal divisions are putting people's jobs and living standards at risk".
May could still face a challenge in Parliament if the opposition Labour Party seeks a confidence vote in the House of Commons over the European Union divorce plan.
"It isn't even clear that all Brexiteers are comfortable with a no deal scenario, which of course will be opposed by the majority of MPs in the House of Commons".
The chancellor, Philip Hammond, had earlier welcomed the no-confidence vote, saying it would "flush out the extremists".