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Boris Johnson has vowed to "keep going" despite two by-election defeats and the resignation by party chair and minister Oliver Dowden.
His party lost the Devon seat of Tiverton and Honiton to the Lib Dems, and Wakefield to Labour.
Mr Johnson described the results as a "disappointment" but said he would continue to address people's concerns.
However, ex-Tory leader Michael Howard said the country would be "better off" if Mr Johnson stepped down.
Speaking to the BBC's World at One, Lord Howard said the members of the cabinet should consider resigning and that party rules should be changed to allow for a confidence vote in the prime minister.
But, Ben Houchen - Conservative mayor of Tees Valley - warned his colleagues against a "knee jerk reaction" and said that holding a leadership contest during a cost of living crisis would make the party look "ridiculous".
Mr Dowden announced his resignation in the early hours of Friday morning following the two by-election defeats.
In a letter to the prime minister, Mr Dowden said supporters were "distressed and disappointed by recent events and I share their feelings".
"We cannot carry on with business as usual," he wrote adding: "Somebody must take responsibility and I have concluded that, in these circumstances, it would not be right for me to remain in office."
Replying to Mr Dowden, Mr Johnson said he understood his disappointment but noted that the government "was elected with a historic mandate just over two years ago to unite and level up".
He is currently in Rwanda attending a Commonwealth heads of government meeting, but has been in touch with senior members of his cabinet.
The prime minister has faced questions over his leadership since stories emerged of Covid rule-breaking in Downing Street during the pandemic.
The by-elections also came amid concerns over rising costs and a squeeze on wages.
The defeats have put renewed pressure on Mr Johnson - who earlier this month faced a confidence vote from his own MPs.
The prime minister won the ballot, but 148 Conservative MPs voted against him, leaving him in a weakened position.
Under current party rules, the prime minister cannot face another confidence vote for one year, but Lord Howard said the 1922 committee should change them to allow another vote.
Cabinet ministers including Culture Secretary Nadine Dorries and Deputy PM Dominic Raab have given the PM their backing.
In a statement, Trade Minister Penny Mordaunt said she was "disappointed" for the party but did not specifically mention support for the PM.
In Tiverton and Honiton - where former MP Neil Parish quit after he was found watching pornography in Parliament - the Lib Dems won with a 30% swing.
The party quashed a Conservative majority of 24,239 - the largest ever to be overturned at a by-election.
Speaking from the Devon constituency, Liberal Democrat leader Sir Ed Davey said the Conservatives were "chaotic" and had "not got the policies for our country".
The Conservatives also suffered a loss in West Yorkshire where Labour secured a 12.7% swing to regain the Wakefield constituency by 4,925 votes.
Labour leader Sir Keir Starmer said the election result was "a clear judgment on a Conservative Party that has run out of energy and ideas".
Labour "is back on the side of working people, winning seats where we lost before, and ready for government," he added.