“The reason I believe that I represent the best opportunity for the U.S. Senate to be under Republican control is because I know the Democrats believe it,” Mr. Tillis said, adding, “They know that when we win the primary, we’re going to take back the Senate and we are going to take this country in a different direction.”

While American Crossroads, the group founded by Mr. Rove, has already run more than $1 million in ads in support of Mr. Tillis, Democrats are taking note of him, too: Outside Democratic groups have already poured roughly $4 million into the race against him.

Most Republican leaders expect Mr. Tillis to finish first in the primary, but state law requires the top finisher to attract at least 40 percent of the vote to avoid a runoff. And while eight candidates are running, Mr. Brannon and Mr. Harris have emerged as those most likely to push Mr. Tillis into a runoff.

Some party leaders privately worry that Mr. Brannon, if he prevails in the primary, could doom their chances in the fall. He was recently found guilty of misleading two investors in a failed start-up company and has been ordered to pay them back more than $450,000, a verdict he is appealing.

He also has a history of remarks that even some in his own party consider provocative: He has praised Jesse Helms, the longtime Republican senator from North Carolina who never renounced racial segregation, as a “modern hero,” and during the 2012 election said a vote for Mr. Romney would “advance tyranny.” Some of the leaders liken him to Todd Akin, the Republican congressman who won the 2012 primary to face Senator Claire McCaskill of Missouri, only to alienate voters with comments suggesting that women who are victims of “legitimate rape” rarely become pregnant.

He waved off the whispers that he is too radical as grumbling from entrenched party leaders. “I think the establishment wasn’t ready for this,” he said. It is not clear how active Mr. Paul will be in Mr. Brannon’s campaign; so far, he has helped him raise money but has not campaigned in the state.

Image At a recent Republican event in Durham, N.C. The notion that the incumbent is weak has brought an influx of challengers. Credit... Jeremy M. Lange for The New York Times

While Mr. Tillis is viewed as the favorite of mainstream Republicans, he is far from moderate: Under his leadership, the legislature passed broad restrictions on voting, rejected the Medicaid expansion provided under President Obama’s health care law and passed an amendment to ban same-sex marriage, among other measures.