The former New Yorkers also tend to favor cosmopolitan political moderates over the kind of hard-core conservative activists who appeal to President Trump. In addition to Mnuchin, their allies have included former chief White House economic adviser Gary Cohn, a former Goldman Sachs banker, and former deputy national security adviser Dina Powell, also a Goldman Sachs executive.

That category does not include Bossie, the former campaign aide and longtime conservative activist, who has protected his own reputation among the president’s aides and advisers, but whose friendship with ex-Trump campaign manager Corey Lewandowski is unsettling for many.

“Bossie would have a pretty good chance of getting the job, if not for them,” a source close to the White House said Tuesday, hours after Bossie appeared on “Fox & Friends” in what some viewed as a kind of audition for the role. (Trump is a well-known viewer of the show.)