Replacing Boris Johnson as prime minister needs to be done “properly and professionally, but quickly”, Education Secretary James Cleverly has told Sky News.

But the cabinet minister ruled himself out as a candidate to become prime minister after Mr Johnson’s resignation effectively fired the starting gun on the Conservative Party leadership race.

Mr Cleverly also disagreed with Sir John Major’s idea of skipping the vote of party activists to speed up the selection process, stressing the need to have leader who was a good campaigner, which Theresa May, having been crowned in the top job, had not been.

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‘No such thing as caretaker PM’

His comments came amid growing pressure from senior Conservatives and political opponents for Mr Johnson to quit Downing Street immediately and not wait for the election of a new leader.

Following the days of turmoil at Westminster, some Tory MPs fear a summer of “chaos” if Mr Johnson remains in office while the leadership contest – which could run for weeks or even months – is held.

Meanwhile, Labour deputy leader Angela Rayner told Sky News the opposition will call a no-confidence vote in Mr Johnson unless he steps down immediately.

Downing Street has dismissed claims Mr Johnson wanted to stay on as prime minister in part because of plans for a wedding party at Chequers, following his low-key marriage last year.

Mr Cleverly said: “I am focusing and the government is focusing on the future, which is the right thing to do and the future is about making sure we have a new prime minister and governmental team in place promptly and in the meantime continue to serve the British people.

“He remains prime minister until a new prime minister is found. That’s how our system works.

“Everyone recognises the incentive of doing that properly and professionally, but quickly so that we can get some certainty and so the new prime minister, whoever that may be, can build their team and make sure they are continuing to focus on what really matters and that’s the people of this country.”

Read more: Who are the frontrunners to be the next PM?

In a warning to would-be candidates, who have been plotting against the prime minister, Mr Cleverly said: “I do think that demonstrating loyalty to others is important if you want people to demonstrate loyalty to you.

“I think that will be one of the criteria that is taken very, very seriously by not just myself but a range of people.”

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The rise and fall of Boris Johnson

He added: “We have a number of people who would make excellent, excellent prime ministers. I think we are spoilt for choice.”

Rejecting the suggestion of a curtailed contest with Tory MPs electing the new leader, Mr Cleverly said: “I think what we saw with Theresa May, that testing a future prime minister’s ability to campaign and campaign across the country is really important.

“Theresa May had many many fantastic attributes, but what we saw in the 2017 general election, she was not a natural campaigner and that was not good for government.

“I think testing the ability of the candidate to reach out beyond the Westminster bubble is incredibly important. So I think it is an important part of the process.”

Ms Rayner said: “We think that Boris Johnson can’t stay another minute in Downing Street. It’s been proven he’s not fit to govern.”

Urging the Conservatives to remove him now, she added: “If they don’t do that we are very clear we will put a motion of no-confidence forward before the summer recess.”

Liberal Democrat leader Sir Ed Davey said: “He should go now, he shouldn’t be allowed to go on.”

Tom Tugendhat became the first Conservative MP to launch a leadership bid following Mr Johnson’s resignation speech.

Writing in the Daily Telegraph, the chairman of the Foreign Affairs Select Committee said his party needs a “clean start”.

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It is understood Transport Secretary Grant Shapps is also considering a Conservative Party leadership bid, as is former health secretary Sajid Javid, who resigned on Tuesday evening.

But Deputy Prime Minister Dominic Raab and Michael Gove have both ruled themselves out of the contest, Sky News has been told.

Foreign Secretary Liz Truss – who was returning early from an international gathering in Indonesia – and Defence Secretary Ben Wallace are both thought to have significant support.

Other ministers considering a run include Attorney General Suella Braverman and Cabinet Office Minister Penny Mordaunt.

Former health secretary Jeremy Hunt, who was runner-up to Mr Johnson in 2019, and arch Brexiteer Steve Baker may also stand.