The Foreign Secretary is said to be concerned over the “direction of travel” with Brexit amid fears that the UK could be shackled to Brussels for much longer in a transition deal.
A Brexit supporter said that a challenge to Mrs May “could come in weeks” before the local elections which many in the party expect to be a disaster for the Conservatives.
Previously, it was thought that nobody would dare challenge Mrs May ahead of an election but the MP suggested that the party had “already written off” the council elections.
The suspicions of Mr Johnson’s cabinet colleagues were heightened this week when he publicly started a row over NHS funding even though it is not part of his brief.
This was followed by a briefing that Sir Graham Brady, the chairman of the 1922 committee which represents Conservative MPs, is only a few letters short of the 48 needed from MPs under party rules for a no confidence vote in Mrs May to trigger a leadership election.
A cabinet minister told the Express: “Things are very unsettled and he (Boris) sees an opportunity.”
The senior minister added: “He possibly thinks it is now or never. The only thing keeping Mrs May in a job is a lack of viable candidates.”
Another Tory MP said: “Boris’ people are going around the tea rooms garnering support. Always when he’s away but it is clear what they are doing.”
The former London Mayor is not the only one believed to be preparing a leadership bid.
There is speculation that Home Secretary Amber Rudd is preparing to run against him in the event of a contest being forced and already has a team in place.
Meanwhile, international development secretary Penny Mordaunt and defence secretary Gavin Williamson are said to be preparing possible bids.
The Daily Express has also been told that at least one senior MP outside the cabinet has put a skeleton leadership team together.
The fevered activity follows Mrs May’s botched reshuffle and concerns over a lull in the Brexit talks.
With discontent growing in the party and fears that they could lose to Jeremy Corbyn at the next election if Mrs May stays in place.
A senior Brexiteer said: “The problem is we think that we could end up with Bino which means Brexit in name only.
“A lot of us do not want a transition period and there is a genuine concern we could end up shackled to the EU for years.
“The other problem is that Theresa May seems determined to keep Philip Hammond as Chancellor and while he is there we will not be able to reap the benefits of Brexit because he is in the pocket of the CBI.”
Another senior MP added that Brexiteers were “running out of patience” with Mrs May because she appears to be taking instructions from senior civil servant Oliver Robbins who has been described as “weak on Brexit”.
In the last few weeks, Mrs May has been publicly criticised by normally loyal MPs – Sir Nicholas Soames and Nick Boles for needing a bigger vision.
Responding to the criticism in an interview yesterdayTHURS, Mrs May insisted she has the vision to lead Britain.
She said: “When I became Prime Minister, I stood on the steps of Downing Street and I set out that I wanted to develop a country that really works for everyone.
“I think one of the challenges we have is those people in our society who feel that globalisation has left them behind, that others have benefitted, but not them.
“We need to ensure that our economies truly do work for everyone.”