Mr Stride, the shadow work and pensions secretary, won 16 votes in the last round of voting, meaning if his supporters go on to support Mr Cleverly too, it would put him within touching distance of the final round.
Meanwhile Robert Jenrick, who has been the frontrunner for most of the race, is expected to secure a slew of senior endorsements ahead of today's vote.
But Tom Tugendhat and Kemi Badenoch's teams insist they have widespread support and the race is "definitely still wide open".
Mr Stride's endorsement came after the shadow home secretary was widely viewed as delivering the speech with the biggest impact at the Conservative party's conference last week.
Mr Stride said: "We must pick a leader who can hit the ground running, take the fight to Labour and make sure that we are ready to win as soon as the opportunity presents itself.
Appeal
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