Reports suggest that the number of Conservative MPs formally expressing no-confidence in Mrs May has reached 40, just a handful short of the figure needed to trigger a party leadership election.
Under Conservative Party rules, a leadership ballot is triggered when 15% of MPs have sent letters to the chairman of the powerful back-bench 1922 committee. The current chairman, Graham Brady, will announce a vote when 48 letters have been collected, although the running total is never publicly confirmed.
Mr Brady has called on colleagues to think carefully before submitting their names in case a crippling leadership challenge is triggered 'by accident'. However, two more letters are expected to be sent in over the weekend, according to the Evening Standard.
The Sun reported that Mr Brady was "ashen faced" at the prospect of an election, although the claims were denied by his allies.
Plotters are understood to be hesitant to trigger a leadership challenge due to the lack of viable candidates to replace Mrs May. A party source told The Scotsman: "If there was someone sounding the trumpet as an obvious alternative, then things might be different, but that isn't the case at the moment."
Boris Johnson, who has continued to challenge government policy, most recently on the NHS, is seen as posing the biggest threat to the Prime Minister, but several other Conservatives inside and outside the cabinet are thought to have also quietly assembled leadership teams.
English local elections in May are seen as a major hurdle, with insiders quoted as saying they represent the "maximum danger point" for Mrs May. Conservatives fear they face significant losses in London, as well as in other cities including Birmingham.