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.