The 2003 contract included a promise that “in no event shall the program be used in any way that denigrates the name, image or reputation of Donald J. Trump,” although it is unclear which provisions of the contract still apply.

It is common Hollywood practice for key personnel on a television show to retain producing credits even after they leave a series, as Mr. Trump did in 2015 when he began his presidential bid. Mr. Trump’s role in the new “Apprentice” season, which finished filming in March, is widely considered a passive one.

His stake in the show’s profits, however, remains active, according to Mr. Trump’s spokeswoman, Hope Hicks, who described Mr. Trump on Friday as “a part owner.” In past seasons, Mr. Trump also received a cut of advertising partnerships with brands like Chrysler, which sponsored segments on the show.

The exact terms of Mr. Trump’s deal for “The New Celebrity Apprentice” remain unclear; Mr. Burnett’s studio, MGM, declined repeated requests this week to elaborate. Mr. Burnett was scheduled to take questions at Friday’s promotional event, but he did not show up.

A financial disclosure form filed by Mr. Trump showed that the company he associates with “The Apprentice,” Trump Productions, earned about $5.9 million from January 2015 to May 2016. As the host, Mr. Trump received appearance fees that he will no longer collect now that Mr. Schwarzenegger is the star.

As NBC reintroduces one of its more popular prime-time shows, the public curiosity over Mr. Trump’s continuing role could prove challenging.

Network executives did not expect Mr. Trump to be president-elect when “The New Celebrity Apprentice” premiered. They also did not anticipate that Mr. Trump would be accused during the campaign by multiple women of sexual assault — allegations similar to ones faced by Mr. Schwarzenegger when he ran for California governor in 2003.