Mr. Kristol said Mr. Walsh’s comfort with the in-your-face format of conservative talk radio makes him a potentially more effective combatant against Mr. Trump than someone like William Weld, the genteel former governor of Massachusetts, whose own primary challenge to the president has gained little traction.
“He has a different appeal than Bill Weld,” Mr. Kristol said. “The fact that he was a Tea Party congressman who voted for Trump in 2016 gives him an ability to speak to Republican primary voters that ‘Never Trumpers’ like me don’t have.”
Tim Murtaugh, a spokesman for Mr. Trump’s campaign, responded to a request for comment with two words about Mr. Walsh’s possible challenge: “Certain failure.”
But Lucy Caldwell, an adviser to Mr. Walsh, said she planned to make an appeal to “traditional campaign consultants who should look beyond fear of losing an R.N.C. contract, and think about why they got into politics in the first place.”
Mr. Trump’s advisers have been mindful for over a year that primary challenges can expose weaknesses in re-election efforts, and can conclude with messy delegate fights on the floor of the Republican National Convention. Led by the former White House political director, Bill Stepien, they have been working to tighten their stranglehold on state parties to make it virtually impossible for any primary challenger to amass delegates.
Some Trump advisers are more concerned about a primary challenge than they publicly let on, as Mr. Trump faces more strain in the job than he has in many months. They are also aware that primary challenges can weaken incumbents in a general election. The most recent example was President George Bush, who faced a primary challenger from the right in 1992 from Patrick J. Buchanan, a former aide to Richard Nixon and a prominent conservative.
In Mr. Trump’s case, there is little to no space to his right for a challenger to occupy. And his lock on Republican voters has remained solid. But primaries can drain energy and resources, particularly when the president is increasingly focusing on distractions, like a feud with his 11-day communications director, Anthony Scaramucci, who has recently turned against his former boss, calling him a “demagogue” and encouraging Republicans to take a stand against him.