Tory Brexiteers are increasingly confident they have enough support to trigger a no-confidence vote in Theresa May as party leader.
If 48 Conservative MPs submit letters to say they no longer support her, a leadership challenge will be launched.
There is no confirmation but sources, including a cabinet minister, have said they believe 48 letters have been sent.
The BBC has also been told the senior backbencher who receives the letters has asked to see the PM on Wednesday.
However, Sir Graham Brady, who chairs the backbench 1922 committee, would make no comment.
Downing Street sources are playing down an imminent move and say they have had no contact from him.
The backbench challenge to May as Conservative leader has been led by those unhappy at the Brexit deal she negotiated, which they argue would keep the UK closely tied to the EU in the long term.
Some Tories who voted Remain in the referendum are unhappy too.
Separately, many opposition MPs have
to call for a Commons vote of no confidence in Mrs May as prime minister, though he has so far resisted.
May spent Tuesday meeting EU leaders and officials in The Hague, Berlin and Brussels, in efforts to salvage her deal, which is widely opposed by MPs of all parties, who voted both Leave and Remain in the EU referendum.
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 Commons.
What have backbenchers said?
So far, 27 Tory MPs have publicly stated they have sent letters saying they have lost confidence in their leader - but speculation has increased that the numbers have risen.
Former Environment Secretary Owen Paterson was among the latest to call for Mrs May to go, saying she had failed to prepare for a no-deal Brexit, tried to bounce her ministers into supporting her and approached negotiations like a "feeble and unworthy" supplicant.
"These mistakes have eroded trust in the government, to the point where I and many others can no longer take the prime minister at her word," he wrote in his letter,
.
"She has repeatedly said 'no deal is better than a bad deal', but it is clear her objective was to secure a deal at any cost."
But former cabinet minister Damian Green, who supports Mrs May, told BBC's Newsnight: "If people have written letters to this degree... this is an act of monumental self-indulgence.
"People outside the Westminster bubble will be looking at this thinking 'we have got a prime minister doing really difficult negotiations at the sharp end of one of the most important decisions this country has taken in 50 years'.
"To undermine a prime minister at this stage seems to me to be wholly wrong."
What has May said?
Asked on Tuesday evening whether she had been told the threshold of 48 letters had been reached, May replied: "No, I have been here in Europe dealing with the issue I have promised Parliament I would be dealing with."
She is due to travel to Dublin on Wednesday after hosting a weekly meeting of her cabinet and facing Prime Minister's Questions.
The PM is then scheduled to attend a summit of European leaders on Thursday at which she is likely to press for changes to elements of her Brexit deal to try to get the support of Parliament.
Speaking on Tuesday, she said there was a "shared determination" among EU leaders to solve the Irish border problem preventing MPs from backing her deal.
European leaders have said they will not renegotiate the text of the withdrawal deal agreed last month, as demanded by many Tory MPs, but are prepared to clarify aspects of it.
What are the rules for any contest?
It is not clear when a contest - if there is one - might take place, but Sir Graham has said in the past it would not be delayed unduly if the threshold were met.