Conservative MPs will vote again on Tuesday as they continue the process of choosing the two candidates for leader who will 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 when the result is announced later, reports the BBC.
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.
Separately, Prime Minister Boris Johnson will hold his last scheduled cabinet meeting on Tuesday ahead of Westminster’s summer recess.
The Conservative Party wants MPs to have chosen the two candidates to replace him before the Commons finishes on Thursday – with the final vote set to take place on Wednesday.
After a summer of campaigning by the two final contenders, grassroots Tories will vote for who should become leader, with the winner to be announced on September 5.
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 Sunak coming out on top having gained 14 votes for a total of 115.
Ms Mordaunt dropped one vote but held onto second place with 82 votes, while Ms Truss gained seven votes to bring her total up to 71.
Ms Badenoch placed fourth with 58 votes, up nine from the previous round.
In a statement, Tugendhat said: “I want to thank my team, colleagues and, most of all, the British people for their support.
“I have been overwhelmed by the response we have received across the country. People are ready for a clean start and our party must deliver on it and put trust back into politics.”
But Ms Mordaunt said: “My vote is steady and I’m grateful to my colleagues for all their support and thrilled to be in second place once more.”
A prominent Sunak supporter said: “That’s a cracking result – Rishi gaining most [votes].”
And a Badenoch campaign source said: “Kemi is pleased to have taken it to the next vote. She has momentum over both Mordaunt and Truss. It’s all to play for and Kemi is in it to win.”
A source within Ms Truss’s campaign told the BBC: “We’ve narrowed the gap to Penny pretty considerably. Story is Penny going backwards. All to play for!”
A Sky News debate scheduled for Tuesday was cancelled after Sunak and Ms Truss declined to take part, and amid concern among senior Tories that airing disagreements in public could damage the party.