Boris Johnson is facing growing calls from senior Tories to hand over to a caretaker prime minister rather than wait for a permanent successor to be elected.
Constitutional experts are clear Mr Johnson is entitled to remain in No 10 until a new party leader, who will be able to form a government, is in place.
But with the leadership election expected to take weeks or even months, some Conservatives warned it is untenable for him to carry on for so long, given the acrimonious way in which he was forced to resign.
Meanwhile, Labour said it will seek a Commons vote of confidence if Mr Johnson does not go, which would mean a general election if the Government is defeated.
Former prime minister Sir John Major said it would be “unwise and may be unsustainable” to allow Mr Johnson to stay on for a prolonged period in which he would continue to be able to wield considerable power.
“Some will argue that his new Cabinet will restrain him. I merely note that his previous Cabinet did not – or could not – do so,” he said.
In a letter to Sir Graham Brady, the chairman of the backbench 1922 Committee, Sir John suggested Deputy Prime Minister Dominic Raab could be installed as acting premier until there is a new leader.
Alternatively, he said, there could be a foreshortened leadership contest, with MPs electing the leader who would then become prime minister – with grassroots members then asked to endorse the result.
There were signs of Sir John’s misgivings being shared by some remaining members of the Cabinet.
Business Secretary Kwasi Kwarteng – who Mr Johnson had reportedly been planning to axe in his next reshuffle – said it is a “depressing state of affairs”.
“We now need a new leader as soon as practicable. Someone who can rebuild trust, heal the country, and set out a new, sensible and consistent economic approach to help families,” he said.
At a meeting of his senior ministers on Thursday, Mr Johnson sought to reassure them he will not seek to introduce new policies and that any major decisions on tax and spending will be for the next prime minister.
However, George Freeman, one of the last ministers to resign before Mr Johnson announced he was stepping down, warned of “chaos” if he is allowed to remain in office over the summer.
“Unfortunately, it’s very difficult to see how Boris Johnson, given the character that he is, is going to be able to govern for three months in quiet humility and contrition.
“That certainly wasn’t the tone of his speech,” he told Times Radio.
“My real worry is the instability will fuel a febrile moment of midsummer madness, where we choose the wrong person in a hurry because of the instability. We can’t afford to do that.”
Former cabinet minister Andrea Leadsom, who twice stood for the leadership, urged Mr Johnson to go for the sake of his reputation.
“I absolutely do not think that it is acceptable that we have a long, drawn-out leadership campaign,” she told BBC Radio 4’s PM programme.
“It is my opinion that Boris would be better to go now and salvage what is a good track record in some really key major events.”
Mr Johnson’s former chief advisor Dominic Cummings tweeted: “Evict TODAY or he’ll cause CARNAGE, even now he’s playing for time & will try to stay. Raab shd be interim PM by evening.”