Conservative MPs are voting in the latest ballot to decide the two leadership candidates to be put to a vote of members.
One of the four remaining contenders – Rishi Sunak, Penny Mordaunt, Liz Truss, and Kemi Badenoch – will be eliminated after Tuesday’s vote.
Mr Sunak looks almost certain to make the final two, with the other three candidates vying for second place.
Senior backbencher Tom Tugendhat was knocked out after Monday’s vote of MPs.
In a series of votes, Conservative MPs are aiming to whittle the field down to two candidates before the Commons breaks for summer on Thursday.
Tory party members will then pick the winner after a summer of campaigning by the two final contenders, with a result announced on 5 September.
Backbench MP Mr Tugendhat was eliminated after receiving 31 votes, down one from his performance in the second round last Thursday.
The other candidates placed in the same order as on Thursday, with former chancellor Mr Sunak coming out on top having gained 14 votes for a total of 115.
Trade Minister Ms Mordaunt dropped one vote but held onto second place with 82 votes, while Foreign Secretary Ms Truss gained seven votes to bring her total up to 71.
Ms Badenoch, a former equalities minister, placed fourth with 58 votes, up nine from the previous round.
Voting to whittle the four contenders down to three will take place on Tuesday afternoon, with a result expected around 15.00 BST.
The vote to decide the final two contenders is scheduled to then take place on Wednesday.
Following Monday’s vote, a source close to Ms Badenoch’s campaign said she had “greater momentum” than her main rivals for the second spot, and the contest was “all to play for”.
The source added that Ms Truss had “only added seven votes” and “hasn’t made a breakthrough”, and Ms Mordaunt had “gone backwards”.
But Ms Mordaunt described her vote as “steady”, and she was “thrilled to be in second place once more”.
A source within Ms Truss’s campaign told the BBC she was “confident” of making it through to the final round.
“MPs who don’t want Rishi Sunak to be PM need to think about who is most likely to beat him in the final two, and can turn the economy round and win an election. Liz is the best bet,” they added.
A Sky News debate scheduled for Tuesday was cancelled after Mr Sunak and Ms Truss declined to take part, and amid concern among senior Tories that airing disagreements in public could damage the party.
In seven weeks’ time, we will have a new prime minister.
For centuries in this country, with a few exceptions, we have collectively been in the habit of getting white men to run the place.
We now know for certain that that isn’t going to happen this time around.
We are now down to the final quartet: former Chancellor Rishi Sunak, Trade Minister Penny Mordaunt, Foreign Secretary Liz Truss and former Equalities Minister Kemi Badenoch.
As the scramble for MPs’ votes continues, Ms Mordaunt has pledged to boost the number of degree apprenticeships available.
In an article for the Daily Telegraph, she has also pledged to replace house building targets for councils with “incentives” to build more, and channel development away from “precious greenfields”.
Meanwhile, Ms Truss has promised that if elected she would increase UK defence spending to 3% of GDP by 2030, to guard against the threats of an “increasingly dangerous world”.
Meanwhile, Boris Johnson held his final Cabinet meeting, telling ministers schools should remain open and transport systems should keep running during the current heatwave.
And he said it was right the UK was pushing for the target of net zero.
At the end of cabinet, ministers gave him six first editions of Winston Churchill’s books on World War Two.