Propelled into the top job to "get Brexit done", Johnson in 2019 won his party's biggest majority in more than 30 years.
But he now faces calls to resign after a series of revelations about parties in the prime ministers' home and office during COVID lockdowns.
Johnson has repeatedly apologised for the parties and said that he was unaware of many of them.
However, he attended what he said he thought was a work event on May 20, 2020 which revellers had been told to "bring their own booze".
To trigger a leadership challenge, 54 of the 360 Conservative MPs in parliament must write letters of no confidence to the chairman of the party's 1922 Committee.
As many as 20 Conservative lawmakers who won their seats at the last national election in 2019 plan to submit letters of no confidence in Johnson, the Telegraph reported. A handful of others have already said they had written such letters.
"Group of 2019 MPs to submit letters to try to hit threshold of 54 to trigger a contest," BBC Political Editor Laura Kuenssberg said on Twitter. "They might hit 54."
An analysis by The Times newspaper showed that 58 Conservative lawmakers had openly criticised the prime minister.
The latest plot against Johnson was cast as the "pork pie plot" because one alleged rebel lawmaker was from Melton, the home of the Melton Mowbray pork pie. Pork pie is also London slang for a lie.
Toppling Johnson would leave the United Kingdom in limbo for months just as the West deals with the Ukraine crisis and the world's fifth largest economy grapples with the inflationary wave triggered by the COVID pandemic, with UK inflation rising to the highest level in nearly 30 years.
Leading rivals within the Conservative Party include Chancellor of the Exchequer Rishi Sunak, 41, and Foreign Secretary Liz Truss, 46.
Johnson on Tuesday denied an accusation by his former adviser that he had lied to parliament about a lockdown party, saying nobody had warned him the "bring your own booze" gathering might contravene COVID-19 rules.
He sidestepped questions about whether he would resign if proven he misled parliament, saying only that he wanted to wait for the outcome of an internal inquiry.
Johnson will address parliament on Wednesday after his Cabinet is expected to approve plans to end the recent restrictions imposed to tackle the spread of COVID-19 in England.
Opposition leaders have accused Johnson of being a serial liar and called on him to step down.
Downing Street lockdown parties - some held when ordinary people could not bid farewell in person to dying relatives - have undermined Johnson's authority.
His own former spokeswoman resigned after she was captured laughing and joking on camera about how to cast a party if asked about it by reporters.
Such was the revelry in Downing Street at one event that staff went to a nearby supermarket to buy a suitcase of alcohol, spilled wine on carpets, and broke a swing used by the prime minister's young son.
The Mirror said staff had even bought a wine fridge for Friday gatherings, events that were regularly observed by Johnson as he walked to his apartment in the building.
Johnson has given a variety of explanations of the parties, ranging from denials that any rules were broken to expressing understanding for the public anger at apparent hypocrisy at the heart of the British state.
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