After more than 40 resignations from within the government, and with many MPs in open revolt, some cabinet ministers went to Downing Street to tell Johnson he needed to go, a source said.
One encouraged him to make a dignified exit by setting his own timetable rather than face a confidence vote.
Dozens have publicly criticised Johnson's integrity after he was forced to apologise for appointing a politician to a role involving pastoral care, and not recalling being briefed the minister had been the subject of complaints about sexual misconduct.
It is the latest crisis to hit his administration after months of scandals and missteps, including a damning report into parties at his official Downing Street residence that broke strict COVID-19 lockdown rules and saw him fined by police.
But despite the clamour for Johnson to resign, James Duddridge, a Conservative MP and close aide of the prime minister, told Sky News the British leader "is buoyant, he is up for a fight" after a meeting with members of his cabinet.
Duddridge said Johnson and the newly appointed finance minister Nadhim Zahawi would set out a joint plan for the economy next week which would include tax cuts.
Johnson sacked Michael Gove, a senior minister who media earlier said had told the British leader he should quit.
On Wednesday night, Secretary of State for Wales Simon Hart joined the list of ministers resigning from cabinet.
"I am not going to step down and the last thing this country needs, frankly, is an election," Johnson earlier told a parliamentary committee.
Johnson said had a mandate from the 2019 national election, which he won with a large majority, and it would not be responsible to walk away from the job in the middle of a cost-of-living crisis and war in Europe.
The prime minister refused to say if he would try to stay in the job even if he lost a confidence vote from his own MPs.
That could come as soon as next week if they agree to change the party's rules, which only allow one such challenge a year. He narrowly won a similar vote last month.
"The prime minister is deluded if he feels he can cling on in the face of collapsed parliamentary support," a senior Conservative politician said on condition of anonymity.
"He is embarrassing the Conservative Party and showing contempt for the electorate."
Culture minister Nadine Dories said she remained behind Johnson and, when asked if others also still backed him, she replied: "Yes, definitely".
The dramatic resignations on Tuesday of his health and finance ministers triggered a growing swell of other ministerial departures and many Conservative lawmakers questioned his fitness to govern.
The ebullient Johnson came to power nearly three years ago, promising to deliver Britain's exit from the European Union and rescue it from the bitter wrangling that followed the 2016 Brexit referendum.
But his administration's combative and often chaotic approach to governing and a series of scandals have exhausted the goodwill of many of his party, while polls show he is no longer popular with the public.
His leadership has been mired in scandals over the last few months, with a committee now investigating whether he lied to parliament about breaches of rules in Downing Street during the pandemic.
There have also been policy U-turns, an ill-fated defence of an MP who broke lobbying rules, and criticism he has not done enough to tackle inflation.
Despite even one-time supporters saying the current crisis could only end with his resignation, Johnson's spokesperson said he was confident of winning another vote by his party.
Johnson has tried to reassert his authority by quickly appointing Zahawi, a rising Conservative star widely praised for the successful rollout of COVID-19 vaccines, as finance minister.
"At some point, we have to conclude that enough is enough. I believe that point is now," said Sajid Javid, in his resignation speech as health minister, with Johnson listening stony-faced.
