Conservative MPs are due to choose the two men who will contest the final stage of their leadership race later.
The remaining field of four candidates will be whittled down to three in another secret ballot, with the result expected at about 13.00 BST.
There will then be a further vote to select the final two, one of whom will be elected leader by party members.
Boris Johnson topped Wednesday’s third ballot with 143 votes, ahead of Jeremy Hunt, Michael Gove and Sajid Javid.
International Development Secretary Rory Stewart was knocked out of the contest on Wednesday evening, after he secured the backing of just 27 Tory MPs.
Mr Johnson is almost certain to make the run-off of 160,000 or so Conservative members who will elect the next Tory leader in a postal ballot, starting next week.
But the race to join the former foreign secretary in the final two remains too close to call.
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Mr Hunt, the foreign secretary, led Mr Gove by just three votes in the third round of voting, with 54 and 51 supporters respectively.
Home Secretary Sajid Javid was in fourth place with 38 votes.
Mr Javid has insisted he will not pull out of the race and it is all to play for. His camp is hoping to attract backers of Rory Stewart.
Mr Stewart has not yet said who he will endorse.
There were reports on Tuesday that he had held talks with Mr Gove about combining forces.
The BBC’s political editor Laura Kuenssberg said there was all kinds of possibilities regarding potential alliances and pacts between the candidates chasing Mr Johnson.
‘Twists and turns’
The leadership campaign has, so far, been dominated by Brexit – with the candidates criticising each others’ plans for getting the UK out of the EU by the 31 October deadline.
Mr Hunt told LBC he believed the EU was prepared to re-open the agreement they reached with Theresa May – rejected three times by MPs – if they found themselves dealing with the “kind of person they could do business with”.
He said the current Withdrawal Agreement containing the controversial backstop – an insurance policy to maintain an open Irish border – was “dead”, and could not get through Parliament.
But he said he was confident the EU would listen to the UK if it came up with technological alternatives to the backstop, which they knew commanded the support of MPs.
“If you give them the kind of prime minister they think they can do business with, who is going to be fair and tough, then my reading of this is they do want to solve this,” he said.
Mr Gove told LBC there were always “twists and turns” in a leadership race, but he was “hopeful” of making it to the final two, given his breadth of support amongst Tory MPs.
While he had not spoken to Mr Stewart since his elimination, he said “having his support would be brilliant”.
Although Mr Johnson had “formidable qualities”, Mr Gove said he felt he would be a better prime minister than his former cabinet colleague and fellow Brexiteer.
“Boris Johnson has communication skills a plenty but when it comes to a forensic examination of Jeremy Corbyn’s programme, and a demolition of him in the House of Commons, I believe I would be better equipped than any of the other candidates.”
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