“The show was a huge success, and the two were discussing future opportunities,” Hope Hicks, a White House spokeswoman, wrote by email on Tuesday.

Mr. Levin, according to two people with direct knowledge of the visit, broached the possibility of Mr. Trump sitting for another interview on his new Fox News series, “Objectified,” a spinoff from last fall’s special. The show, which is expected to make its debut in September, is to feature interviews with celebrities who describe cherished objects in their lives.

One of Mr. Levin’s ideal guests is Tom Brady, the New England Patriots quarterback and a friend of Mr. Trump, and Mr. Levin planned to ask the president if he would help secure the athlete’s participation, according to one of the people who described the visit.