Nearly two dozen other House Republicans — about half of them politically vulnerable and many in districts with large Hispanic populations — have followed suit, affixing their signatures to a petition that is just a few votes short of forcing Mr. Ryan to act. If it works, it will push the party into a divisive, and politically risky, election year debate on immigration. To avoid such a showdown, the speaker has scheduled a two-hour meeting on immigration with his rank and file when lawmakers return to Washington next week.

“There have been some critics who say that this could cost us our majority,” said Mr. Denham, who is facing a crowded field of five Democrats and one other Republican in Tuesday’s unusual “two party” primary election here. “My concern is if we do nothing, it could cost us our majority. So yes, it’s risky. But it’s the right thing to do.”

From Miami to the Rocky Mountain West to Texas and to California, the petition’s signatories form a renegade band that is not only making life difficult for Mr. Ryan but also standing up to the conservative Freedom Caucus, which so often dictates policy in the House. Of the 23 Republicans on the petition, 11 are in districts that are clearly in play. Another five have announced their retirements or have already left the House.