Rebel Conservative MPs say they expect a no confidence vote in Boris Johnson to be held in the coming days.
Letters expressing no confidence in the prime minister must be submitted by 54 Tory MPs to trigger a challenge.
Data collated by Sky News shows that 28 Conservatives have publicly called for Mr Johnson to resign, with several more questioning his position.
Business minister Paul Scully also said on Sunday evening there “may well” be a confidence vote, but predicted the prime minister would “face down” that challenge.
The process to trigger a ballot is anonymous, and only the senior MP Sir Graham Brady – who chairs the 1922 Committee of Conservatives – knows who has sent in a letter.
One rebel MP told Sky News they suspected the vote would happen this week, but added it was hard to know precisely because this was an “organic and completely unorganised” show of concern from backbenchers.
Earlier this year, suggestions of an imminent leadership challenge failed to materialise and one MP who has sent a letter cautioned that “anything could happen” in the coming days.
There have been suggestions the ballot should be pushed back to later in the month to maximise the chances of removing the prime minister.
‘Boris is going nowhere’
One senior MP who has called for Mr Johnson to resign said having a vote now would be “daft” and “too risky”.
If a contest is triggered, 180 Conservative MPs would need to vote against the PM for him to be removed.
Allies of Mr Johnson said he would win the vote whenever it arrived, meaning one year of immunity from a further challenge, according to party rules.
“After the 12 months, we will be very close to a general election, so probably a whole lot of reluctance to remove him then…Boris is going nowhere and will be leading us into the next general election,” said one supportive ex-minister.
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Downing Street will try to shift the focus onto domestic policy this week, with a series of announcements about the NHS and a potential speech on housebuilding.
However, a poll on Sunday predicted more trouble for the prime minister, forecasting a heavy loss in the Wakefield by-election later this month.
“The main reason that voters give is that Boris Johnson has covered up partygate and then lied about it,” said James Johnson, a pollster at JL Partners.
“The second-biggest reason is Boris Johnson is out of touch with working class people.”
Activists have also criticised the prime minister, with the chair of the Grassroots Conservatives group Ed Costelloe telling Sky News that Mr Johnson should “go with dignity” in the coming months.
“If he stays as the prime minister, the party are very unlikely to win the next election, which would mean some sort of coalition of Labour, Lib Dems and SNP,” said Mr Costelloe.