Mr Cleverly overtook Mr Jenrick to come out on top with 39 of his colleagues backing him in today’s vote.
Mr Jenrick followed with 31 votes, then 30 for Ms Badenoch, and 20 for Mr Tugendhat.
A further vote by MPs on Wednesday will reduce the field to a final two, who will then go to a ballot of Tory members next month.
Mr Tugendhat thanked those who backed him in a post on X, adding: “Your energy, your ideas and your support have shown a vision of what our party could become.
“Our campaign has ended but our commitment to our country continues.”
Mr Tugendhat served as security minister under Mr Sunak’s premiership and before becoming an MP was a member of the armed forces – a fact he has honed in on to pitch himself as the candidate of public service.
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Seen as being from the moderate, centrist wing of the party, he was equal third with former home secretary Mr Cleverly in the last round of voting, when Mel Stride was knocked out.
Mr Cleverly saw a surge in support after the Conservative Party conference last week, when he gave a well-received speech urging party members to be “more normal” and “sell Conservatism with a smile”.
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In a post on social media after the vote, he said he was “grateful” to his colleagues who backed him and “pleased to be through to the next round”.
He added: “The job’s not finished. I’m excited to keep spreading our positive Conservative message.”
Mr Cleverly is likely to make the final round as the “One Nation” wing’s candidate, while Mr Jenrick and Ms Badenoch are seen as battling for the support of the right of the party.
They were previously considered to be the frontrunners, coming in first and second place respectively in the earlier two rounds of voting.
They both faced criticism for comments they made during the party conference – Mr Jenrick for claiming the SAS is being forced to kill rather than capture terrorists because the “European Court will set them free”, something many of his colleagues disputed.
A spokesperson for her campaign said the party’s right-wing “needs to coalesce around Kemi” given the fall in Mr Jenrick’s support, adding she “can reach across and unify the party, has the star quality to cut through in opposition, and is indisputably the members’ choice for leader”.
Mr Jenrick’s campaign said the former immigration minister, who wants to leave the European Convention on Human Rights, was “in prime position to make the final two”.
A source said: “MPs want seriousness and competence. That’s why he’s won support from across the party so far – from Danny Kruger on the right to Vicky Atkins on the left.”
Polling suggests the Conservative membership favours Ms Badenoch over Mr Jenrick and Mr Cleverly, but that the gap has narrowed since the party conference.
Whoever wins on 2 November, when the final result is declared, they face the task of rebuilding the party after their worst-ever general election defeat put them in opposition for the first time in 14 years.
Wyoming has become the latest US state to propose a bill for a Strategic Bitcoin Reserve, just days before Donald Trump’s US presidential inauguration.
No doubt Poland’s prime minister, Donald Tusk, was trying to be friendly. After all, as Sir Keir said, they share a passion for Arsenal Football Club.
But when Mr Tusk declared at their joint news conference in Warsaw that his dream was “instead of a Brexit, we will have a Breturn”, Sir Keir visibly cringed.
Was it an ambush? Not quite. But it was certainly awkward for the UK prime minister. He stood stiffly and didn’t respond, not once uttering the word “Brexit”.
Mr Tusk, however, has form for bemoaning Brexit. He was, after all, the president of the European Council when the UK voted to leave the EU in 2016.
He might now be in his second spell as Poland’s PM, but his five years at the EU make him the ultimate Brussels insider, who’s never made any attempt to hide his feelings on Brexit.
Prior to the UK referendum, in September 2015, he said Brexit “could be the beginning of the destruction of not only the EU but also of western political civilisation in its entirety”.
Standing alongside Sir Keir, he revealed that “for obvious reasons” they discussed co-operation between the UK and the EU. He recalled that his emotional reaction to the referendum in 2016 was “I already miss you”.
He went on: “This is not just about emotions and sentiments – I am aware this is a dream of mine, that instead of a Brexit we will have a Breturn.
“Perhaps I’m labouring under an illusion. I’d rather be an optimist and harbour these dreams in my heart – sometimes they come true in politics.”
A dream? Or a calculated move? As a Brussels insider, was Mr Tusk speaking for the EU as a whole? Was he doing Brussels’ bidding?
He may have returned to lead his homeland, but he remains a key player in Brussels.
On becoming Poland’s PM in 2023, he ended a dispute with Brussels which unlocked billions of frozen EU funds for his country.
He also orchestrated the return of his centre-right ally Ursula von der Leyen as European Commission president.
And Poland has just taken over the rotating presidency of the EU, which means Mr Tusk will be hugely influential once again, chairing meetings and setting agendas.
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Poland is back in the European mainstream. It’s where Mr Tusk would like the UK to be as well.
It’s where, privately, Sir Keir would like the UK to be. It’s just that with Reform UK almost neck and neck with Labour in the polls, he daren’t say so.