Allies of James Cleverly believe his “likability factor” will now hand him a serious chance of reaching the final runoff to replace Rishi Sunak as Tory leader, as MPs prepare to eliminate another candidate from the contest this week. The next vote of MPs will take place on Tuesday to whittle the field down to the four contenders who will have the chance to make their case at the party’s conference this autumn. The race is seen by most as being wide open after a close result in the first round last week. While Robert Jenrick and Kemi Badenoch emerged first and second after the first round of votes among their colleagues, shadow home secretary Cleverly was only a single vote behind Badenoch. Some figures in the party now believe he has the ability to draw support from both its right and the liberal wing. “It’s dawning on people that probably only James can deliver unification,” said a former cabinet minister. Many MPs from the One Nation group of liberal Tories had been backing Tom Tugendhat, who finished in fourth place last week. Cleverly’s allies are now wooing them, as well as seeing if they can win over Badenoch supporters and those who backed Priti Patel, who was eliminated in Wednesday’s first round of voting. They believe that with Tory MPs in charge of this phase of the contest, Cleverly’s ability to charm colleagues and acquire few political enemies during a lengthy ministerial career is his underrated strength. Some MPs voting for him said his “likability” meant they were increasingly confident he would make the final two put forward to a vote among the Tory party membership. Jenrick, a former immigration minister, has now emerged as the frontrunner. However, one veteran Tory said he was still unclear on whether Jenrick was sincere in his claims that he wanted to take the party farther to the right on immigration, or whether he was “doing a Starmer” – a reference to the prime minister’s leftwing Labour leadership campaign, before tacking to the centre as the party’s leader. While shadow work and pensions secretary Mel Stride is considered most likely to lose out in Tuesday’s vote, shadow security minister Tugendhat is also under pressure. He could find himself out of the race if Cleverly manages to convince any backers that he should now be the chosen candidate from the centre. With the race so close, there have already been claims of dirty tricks. Allies of Badenoch have accused Jenrick supporters of “lending” votes to Cleverly as a way of giving him momentum and pushing Badenoch out of the race. The claims have been denied by figures close to Jenrick.