WASHINGTON—A bid by House Majority Whip Tom Emmer (R., Minn.) to serve as the House Republicans’ pick to be speaker will test whether the strong ties he built recruiting candidates and counting votes will overcome doubts from some anti-establishment lawmakers aligned with former President Donald Trump.

Candidates are expected to pitch their colleagues at a forum on Monday evening ahead of an internal vote to designate a new Republican speaker nominee as soon as Tuesday morning. Beyond winning the GOP ballot, the speaker nominee will face the uphill battle to unite almost all Republicans to have a chance of winning the House vote, given Republicans’ narrow 221-212 majority.