WASHINGTON—House Republicans began openly maneuvering ahead of next week’s House GOP leadership elections after months of below-the-radar jockeying as GOP lawmakers braced to return to the chamber’s minority for the first time since 2011.

For now, House Majority Leader Kevin McCarthy (R., Calif.) remained the front-runner for the minority leader post, after formally announcing he would run for the role Wednesday. Hours earlier, Rep. Jim Jordan of Ohio, an influential conservative, said he intended to challenge Mr. McCarthy...