Though Mr. Prince carries baggage, Republicans have privately said that a primary challenge against a lawmaker like Mr. Barrasso is the kind they fear most: an out-of-the-blue run by a renegade from the right against a senator whose sin is not a lack of conservative credentials, but an association with Mr. McConnell and other party leaders.
Those anxieties became all the more serious late last month when Roy Moore, a conservative firebrand, defeated Senator Luther Strange, a McConnell ally, in an Alabama Republican primary. Allies of Mr. McConnell spent tens of millions of dollars defending Mr. Strange, but Mr. Moore won by nine percentage points.
The Prince campaign, if it materializes, would be just the kind of race that Mr. Bannon hopes to replicate across the country. With financing from the Mercers and their network of other conservative donors, Mr. Bannon is looking to build a political coalition that recruits people to run against Republican incumbents from Maine to Montana.
He has set his sights most immediately on states like Arizona, Nevada and Mississippi, where he believes a more populist, maverick-style conservative could threaten a sitting Republican senator. But he is also eyeing candidates to run for the Senate in Tennessee, where the retirement of Bob Corker leaves an opening, and in Nebraska, where he believes that Senator Deb Fisher, a first-term Republican, is vulnerable.
In meeting with potential candidates, Mr. Bannon has emphasized one credential above all else, people who have spoken with him said. He wants to ensure they will commit to voting against Mr. McConnell for Republican leader.
Mr. Prince, 48, has strong ties to the Trump administration. He served as an informal adviser during the transition, and he is the brother of Betsy DeVos, the education secretary. He has told his sister that he would like to run against Mr. Barrasso, a person with knowledge of the conversation said.
In 1997, Mr. Prince founded Blackwater as a private, for-profit force to aid the military, and he is wealthy enough to self-finance his race. For months this year, Mr. Prince — with Mr. Bannon’s support — pushed a plan to replace soldiers with contractors in Afghanistan. The proposal, which would have radically changed the way the fight in that country is conducted, was vehemently opposed by the national security adviser, Lt. Gen. H. R. McMaster, and the defense secretary, Jim Mattis.
Beyond his support of the use of contractors in war zones, Mr. Prince’s views on political issues are less widely known. He has been described as a libertarian. In 1992, he supported Pat Buchanan’s run for president.
Mr. Prince is emblematic of the type of parallel government apparatus that Mr. Bannon built inside the administration during his time as Mr. Trump’s chief strategist. Mr. Bannon, with help from the Mercer family, is now trying to build the same kind of parallel structure inside the Republican Party. And Mr. Prince is not the only candidate on his wish list.
Mr. Bannon is expected to throw his support behind Chris McDaniel, a conservative state senator from Mississippi who is considering a primary challenge to United States Senator Roger Wicker, who has served since 2007 and is close to Republican leaders.
Mr. Bannon is also hoping to persuade Ann LePage, the wife of Maine’s outspoken governor, Paul LePage, to run for the Republican nomination to challenge Senator Angus King, an independent who is up for re-election in 2018.