The Prime Minister has refused to name a date for her resignation, amid growing pressure from Tory MPs to make way for a new leader.
Sir Graham Brady, chair of the 1922 Committee, met with the Prime Minister privately on Tuesday, and has since made it clear she had not offered him any further clarity regarding her future in the role.
He suggested that Mrs May could make a final effort to get her Brexit deal through Parliament before the European elections on May 23.
Yesterday, Conservative MP Andrea Jenkyns asked Mrs May directly to "step aside" in a heated Prime Minister's Questions.
Brexiteer Ms Jenkyns told Mrs May she had "failed" in EU withdrawal negotiations and forfeited the trust of the public, but the Prime Minister dismissed her suggestion.
Mrs May retorted: "This is not an issue about me and it's not an issue about her.
"If it were an issue about me and the way I vote, we would already have left the European Union."
There is speculation that the Prime Minister will seek to hang on until the annual party conference in the autumn, after her effective deputy, David Lidington, suggested he hoped a Brexit deal could be agreed by July.
Nigel Evans, the executive secretary of the 1922, warned she would need to give MPs some greater certainty before then.