If 48 Conservative MPs write to senior backbencher Sir Graham Brady that they no longer have confidence in her, a leadership challenge will be launched.
Several sources, including a cabinet minister, have told the BBC they believe 48 letters have been submitted.
But there is no official confirmation that threshold has been reached.
The BBC has also been told that senior backbencher Sir Graham Brady has asked to see the PM on Wednesday.
Sir Graham, the chair of the backbench 1922 committee to whom MPs have to address their letters, would make no comment.
Downing Street sources are playing down an imminent move and say they have had no contact from him.
Mrs May has spent the day meeting EU leaders and officials in The Hague, Berlin and Brussels, in efforts to salvage her Brexit deal - which faces major opposition in Parliament.
Her decision to delay the vote on the terms of the UK's withdrawal from the EU, which had been due to take place on Tuesday, has caused anger across the party.
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