Mrs May was hit by the resignation of Andrea Leadsom, Leader of the House of Commons, on Thursday evening, who quit saying in a letter: "I no longer believe that our approach will deliver on the referendum result".
She also cited a breakdown of collective responsibility as another of her reasons for stepping down.
Speaking following her resignation Mrs Leadsom told reporters she quit because she couldn't, as leader of the Commons, "announce a bill that I just think has elements I cannot support that aren't Brexit".
Mrs Leadsom is the 36th minister to quit during Mrs May's time in Downing Street.
She added: "I have been determined to deliver Brexit and I'm just worried that this bill, with it's new elements in it, would not do that."
Mrs Leadsom said she was "proud" to have served in Theresa May's Government since 2016, and had stayed in the Cabinet to "shape and fight for Brexit" despite some "uncomfortable compromises along the way".
But she said: "I no longer believe that our approach will deliver on the referendum result."
Mrs Leadsom said she did not believe that the UK will be "truly sovereign" through the deal proposed, and said a second referendum would be "dangerously divisive".
Mrs Leadsom's resignation piles significant pressure on Mrs May to do the same. However, if she doesn't she could be ousted by the 1922 Committee.
The committee met on Wednesday evening to discuss changing party rules - paving the way for a no confidence vote in the PM.
The Prime Minister will meet Sir Graham Brady, who leads the 1922 Committee, on Friday, for a meeting which could seal her fate.
She has previously agreed to set out the timetable for the contest to replace her after a crunch vote on her Brexit deal, widely expected on June 7.
That deadline appears to have been brought forward with the announcement she will meet Sir Graham.
It had earlier been reported that a decision on rule changes had not been made however ITV News Political Correspondent Paul Brand understands a secret vote on the matter was held.
He told ITV News at Ten: "I understand that was a secret ballot - MPs put their votes into sealed envelopes and those envelopes will only be opened on Friday if the PM refused to resign."
In a sign of unrest at the highest levels of the Tory Party, a series of Cabinet ministers asked for meetings with Mrs May to raise their concerns about the WAB which would put the Prime Minister’s Brexit plan into law.
Downing Street sources said it was possible that the PM could meet Foreign Secretary Jeremy Hunt on Thursday, but there was no confirmation that an audience would be granted for other ministers with reservations about the plan.
Home Secretary Sajid Javid has asked for a meeting to discuss his concerns about the prospect of a second referendum, after Mrs May revealed she would grant MPs a vote on whether the Brexit deal should be put to the public.
Scottish Secretary David Mundell also has particular concerns because he fears the prospect of another referendum could be “exploited” by the SNP to call for a fresh independence vote.
Mrs Leadsom, explaining the timing of her decision to resign on the eve of polling day in the European elections, said: "I considered carefully the timing of this decision, but I cannot fulfil my duty as Leader of the House tomorrow, to announce a Bill with new elements that I fundamentally oppose.
"I fully respect the integrity, resolution and determination that you have shown during your time as Prime Minister.
"No one has wanted you to succeed more than I have, but I do now urge you to make the right decisions in the interests of the country, this Government and our party."
Without a leader of the House Mrs May will not be unable to table her newly revised Withdrawal Agreement Bill, however it was already expected to be defeated.
ITV News Political Correspondent Paul Brand says Mrs Leadsom's resignation means Mrs May will have "to resign by time MPs return from recess, at the very latest".
How did Theresa May react?
Mrs May said she was "sorry" to receive Mrs Leadsom's resignation letter, writing in reply that she was "grateful for the support you have given over the last three years" in working to deliver Brexit.
But the Prime Minister said she disagreed with the assessment Mrs Leadsom gave about the Government's approach now.
She said: "I do not agree with you that the deal which we have negotiated with the European Union means that the United Kingdom will not become a sovereign country.
"If the deal is passed, the UK will leave the European Union. We will leave its Common Agricultural Policy and its Common Fisheries Policy.
"We will end the jurisdiction of the European Court of Justice and make our own laws in Westminster and our devolved parliaments.
"We will stop sending vast annual sums of taxpayers' money to the EU, and spend it on our priorities instead - such as our National Health Service and our children's schools."
Mrs May said she agreed that a second referendum would be divisive, but said the Government was not proposing to hold one.
She wrote: "I continue to believe that the arguments against a second referendum are strong and compelling, and will continue to oppose one."
The PM said she also did not recognise what Mrs Leadsom said about decision-making in Government.
"Indeed, through your own work chairing the Parliamentary Business and Legislation committee of Cabinet, you have been an important part of ensuring that Brexit-related legislative proposals are properly scrutinised, and the whole Cabinet have listened to the assessments you have given at Cabinet meetings," Mrs May said.
"I have valued the frank and productive discussions we have had over the last three years.
"As you say, there are important elections tomorrow, and many Conservatives are working in a challenging environment to support our excellent candidates. I am sorry to lose someone of your passion, drive and sincerity from HM Government in this way."
Will anyone else resign?
Rumours had been circling in Westminster on Wednesday that Mrs Leadsom could quit after she and a number of other Brexit-supporting colleagues in the so-called Pizza Club were absent for the start of Prime Minister’s Questions.
Mrs Leadsom’s resignation will spark fears in Downing Street that others could follow suit, and eyes will be firmly on the likes of Brexiteer Cabinet ministers including Liam Fox, Penny Mordaunt, Michael Gove, and Chris Grayling.
What happens to the Withdrawal Agreement Bill now?
With no Commons Leader to introduce the Business Statement to MPs on Thursday, it is unclear who will outline what will happen in the House in the first week of June.
The PM could appoint a new Commons Leader overnight ahead of the statement, or ask another minister to step in.
It will be up to the Government to decide if they still want to include the WAB in the business for the week of June 3.