Mr. Nehlen welcomes the speculation. In an interview on Wednesday, he called Dave Brat, the economics professor who defeated Mr. Cantor, “an inspiration” and a hero who had helped inspire his own campaign.

“He was a different David to a different Goliath,” Mr. Nehlen said.

On the ground here in southeastern Wisconsin in a district dotted with factories and cornfields, there are few signs that the news media glow surrounding Mr. Nehlen has rubbed off on voters. As he crisscrossed the district on Wednesday, holding news conferences and at least one rally, there were often more members of the news media and his campaign staff on hand than actual voters. Public polling shows that Mr. Ryan is vastly more popular among Republican voters in his congressional district than Mr. Trump.

“This is one of those campaigns that looks big on social media and in certain conservative media, but when you get down on the ground in Racine, in Kenosha and Janesville, it’s just not there,” said Charlie Sykes, a conservative radio host who holds sway among Republicans here. “The grass-roots Republican activists are not divided on Paul Ryan.”

On Monday, Mr. Nehlen went so far as to say he would be open to deporting all Muslims from the country because, he alleged, they may hold Shariah law above the Constitution and are urged to lie by their religion. “I’m suggesting we have a discussion about it, that’s for sure,’’ he said on a Chicago radio show, adding: “We should be monitoring every mosque. We should be monitoring all social media.”

Mr. Nehlen, who is far more conservative than many voters in Mr. Ryan’s district, has still managed to create headaches for the speaker, who has found himself fending off an attack from an opponent to whom he had given little thought, but who has captured the energy and resources of anti-establishment conservatives across the country.