The hunt is on to find a suitable conservative candidate who can beat Mr. Ryan, who received 94 percent of the vote in the 2014 primary in his district, which sits against the Illinois border in the southeastern part of the state.

“There are people who are seriously looking for that,” said James Murphy, founder of Green Bay Tea Party. “There is a sympathetic ear to having someone beat him.”

The Ryan campaign team in Wisconsin has shrugged off the threats, treating them as mere talk in the absence of a viable candidate to mount a challenge in the district.