Boris Johnson is to stand down as Conservative Party leader, but intends to carry on as prime minister until the autumn.
He plans to stay in Downing Street until a new Tory leader has been elected to replace him as PM.
And he has begun appointing new ministers to replace the ones that quit in protest at his leadership.
But some Tory MPs are urging him to leave as soon as possible to avoid government paralysis.
Former minister Sir Bob Neill told MPs there was a "serious question mark" over how long a "caretaker" prime minister could stay in place.
"Might it not be in everybody's interest to speed up the transition as much as possible?" he added.
Sir Keir Starmer said that if the Conservative Party did not "get rid" of Mr Johnson immediately then Labour would bring a vote of no confidence "in the national interest".
"We can't go on with this prime minister clinging on for months," said the Labour leader, adding: "He needs to go completely... he's inflicted lies, fraud, and chaos in the country."
A vote of no confidence would be held in Parliament - if the government lost the vote that could lead to a general election, but this would require a significant rebellion from Conservative MPs to back a Labour motion.
It follows a dramatic 48 hours which saw dozens of ministers - including chancellor Rishi Sunak - resigning in protest at his leadership.
Mr Sunak's replacement as chancellor Nadhim Zahawi was among the ministers urging the PM to quit.
Mr Johnson resisted the calls until Thursday morning, when it became clear that he had lost the confidence of his MPs and that the government could no longer function.
Attorney General Suella Braverman and leading backbencher Steve Baker are the first Tory MPs to declare a leadership bid, with others expected to follow.
Less than three years ago, Mr Johnson won an historic landslide victory in a general election - but he has been dogged by controversy in recent months, including a fine for breaking his own lockdown laws.
The revolt this week was triggered by revelations about the prime minister's handling of sexual misconduct allegations against former Deputy Chief Whip Chris Pincher.
BBC political editor Chris Mason said Graham Brady, the chairman of the 1922 Committee of backbench Tory MPs, has met the prime minister to tell him he has lost the confidence of the party.
I'm fairly sure the cowards in Toryland aren't going to go along with Starmer's no confidence vote this time because they'd actually lose a general election, so expect months of infighting about a new Tory leader who would then become PM. Some 18 Tories want to be the next Boris.
Still, it's possible in a world where enough of the government has resigned to the point of "no longer being a going concern" as the Brits say.
Now the race begins to see how fast Boris Bad Enough can get gone good enough.