According to the Daily Mail, the Prime Minister wants to hold another vote on her deal in the next 10 days, in an attempt to stop a potential bloodbath in the upcoming EU elections.
Officers of the Tory backbench 1922 Committee met in Westminster on Tuesday in an attempt to pile pressure on Mrs May to name the date of her departure.
Senior Conservatives are expected to hold further discussions today on whether to change the party rules to enable an early leadership challenge.
Ahead of Tuesday's talks, the committee chairman Sir Graham Brady met privately with Mrs May, when he is reported to have told her MPs want her to announce when she is going.
Under current party rules, MPs cannot mount a fresh leadership challenge until 12 months after last December's failed attempt.
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However, amid growing frustration over the latest delays to Brexit, some MPs now want to the rules to be rewritten to allow another challenge as early as June.
One member of the executive, joint executive secretary Nigel Evans, has called publicly for Mrs May to go "as soon as possible”.
He said: “I believe the only way we're going to break this impasse properly is if we have fresh leadership of the Conservative Party. If there was an announcement today by the Prime Minister then of course we could start the process straight away.”
Mr Evans added: “I have never said this before. I did say to Theresa May 'Do not breathe fresh life into Nigel Farage', and that's exactly what she has done. He has gone from his death bed to the Olympics, because Brexit hasn't happened.”
However other members were reported to have pushed back at Tuesday's meeting, questioning what a fresh leadership contest at the current time would achieve.
Mrs May has already sought to buy time, promising Tory MPs last month that she would go once she has delivered Brexit.
But following the latest extension to the Article 50 withdrawal process there is growing impatience among her critics.
They fear the Tories will suffer heavy losses to Mr Farage's Brexit Party if - as now seems likely - the UK is forced to go ahead with voting in the European elections on May 23.
Among the potential candidates for Mr Farage's new outfit will be former shadow home secretary Ann Widdecombe, who told the Daily Express: "The public needs to send a very clear message and that is we expect the vote to be respected so just get on with the job of getting us out of the EU.”
Meanwhile, cross-party talks between the Government and Labour aimed at forging a common way forward are continuing amid recriminations at the slow pace of progress.
After talks resumed in Whitehall on Tuesday following the Easter break, Chancellor Philip Hammond is due to have a further meeting with his Labour counterpart John McDonnell on Wednesday.
However the Prime Minister's official spokesman said that while the discussions with Labour were "serious" they were proving "difficult" in some areas, and that progress was needed "urgently" to enable Britain to leave the EU as soon as possible.
Labour leader Jeremy Corbyn, however, put the blame for lack of progress on the Government's refusal to shift on its "red lines”.
He said: ”We'll continue putting our case but quite honestly there's got to be change in the Government's approach.
"They cannot keep on just regurgitating what has already been emphatically rejected three times by Parliament, there's got to be a change."