Boris Johnson faces a growing revolt over suggestions he may cling on as a caretaker Prime Minister until as late as October.
It came as three prominent Conservatives Tom Tugendhat, Steve Baker and Suella Braverman launched their own leadership campaigns.
Senior Tory MP Tom Tugendhat was the latest to announce he will run to replace Boris Johnson.
Writing in The Telegraph, Mr Tugendhat said: “I have served before - in the military, and now in Parliament. Now I hope to answer the call once again as prime minister.
“It’s time for a clean start. It’s time for renewal.”
The PM resigned at 12.30pm after 60 members of his government, including five cabinet ministers, quit.
He said he would stay in Number 10 until a new leader had been found and chaired a meeting of his placeholder cabinet at 3pm.
It is believed he could stay in charge in Downing Street until the Conservative Party conference later this year.
But the decision was blasted by several senior Tories, with some suggesting deputy PM Dominic Raab should take over until Mr Johnson's successor is found.
Bromley MP Sir Bob Neill said: "There's a serious question as to how long a caretaker Prime Minister can remain in place when there is real concern on whether the Government can be fully and effectively manned. We should speed up the transition as much as possible."
George Freeman, who quit as science minister on Thursday morning, said: “Now PM has finally done the decent thing he needs to hand in the seals of office, apologise to Her Majesty, allow her to appoint a Caretaker under whom Ministers can serve, so the Conservative Party can choose a new leader properly."
Sir John Major said Mr Raab should serve as acting prime minister until a new leader is elected.
The former Conservative Prime Minister, who took over when Margaret Thatcher was ousted by her party, said it was “unwise and may be unsustainable” for the PM to stay in place.
In a letter to Tory 1922 Committee chairman Sir Graham Brady, he said: "The proposal for the Prime Minister to remain in office - for up to three months - having lost the support of his Cabinet, his Government and his parliamentary party is unwise, and may be unsustainable.
"In such a circumstance the Prime Minister maintains the power of patronage and, of even greater concern, the power to make decisions which will affect the lives of those within all four nations of the United Kingdom and further afield.
"Some will argue that his new Cabinet will restrain him. I merely note that his previous Cabinet did not - or could not - do so."
Conservative MP for North West Leicestershire Andrew Bridgen urged Sir Graham Brady to shorten the process to replace Mr Johnson.
He told the PA news agency: “I have spoken with Sir Graham Brady and urged him to truncate, wherever possible, the leadership election process.
“I think we will have two candidates out of the parliamentary party to present the membership before recess. And that will be no more than three or four weeks.
“So by the end of August we will have a new leader of the Conservative Party. So Boris Johnson’s estimate that he’ll still be Prime Minister in October is wildly inaccurate.”
The PM has replaced the cabinet members who left his administration but still has more than a dozen junior minister positions to fill.
Labour leader Sir Keir Starmer also warned Mr Johnson should not be allowed to linger in Downing Street and threatened to call a no confidence vote in the Commons if he refused to hand over the reins to a caretaker premier.
Sir Robert Buckland defended his decision to take a role in Mr Johnson's new cabinet and said he was "here to help".
The new Welsh Secretary added Mr Johnson would not have the authority to do "new" things in government.
He said: "I felt it was right that I did that now. The issue about the Prime Minister and his character has been settled. He no longer has the confidence of the Conservative Party. He is resigning. That is right. But the business of government goes on. I'm here to help."
Former Prime Minister Theresa May was asked if she would be prepared to act as a caretaker but said: "I don't think there's going to be a caretaker Prime Minister in the sense of somebody else coming into that role."
Mr Johnson vowed in his resignation speech to serve until the new leader was in place.
He said: “I’ve agreed with Sir Graham Brady, the chairman of our backbench MPs, that the process of choosing that new leader should begin now and the timetable will be announced next week.
“And I’ve today appointed a Cabinet to serve, as I will, until a new leader is in place.”
He added: “Above all, I want to thank you, the British public, for the immense privilege that you have given me and I want you to know that from now on until the new prime minister is in place, your interests will be served and the Government of the country will carry on.”
