Even as Elon Musk has drawn enormous attention for his slash-and-burn assault on the federal bureaucracy and his partnership with President Trump in the opening weeks of the new administration, quieter efforts are underway to position Vice President JD Vance as Mr. Trump’s eventual successor.

Mr. Vance is expected to lead some of the legislative battles over Mr. Trump’s agenda, as he is viewed inside the White House as the administration’s chief lobbyist in the Senate.

He has also been assigned to help oversee the potential sale of TikTok — a reward, according to one person briefed on the matter, for working on a plan, including one night until after 2 a.m. before the inauguration, that postponed a ban on the app that was set to take effect.

Mr. Vance and those around him have also applied a 2028 lens in early jockeying within the administration.

When Mr. Trump was toying with the idea of a cabinet post for Gov. Ron DeSantis of Florida, three top advisers pulled Mr. Vance aside and argued that he should oppose such an appointment for a possible rival for the 2028 Republican presidential nomination, according to three people briefed on the private discussions who insisted on anonymity. Mr. Vance disagreed, saying he saw some political upside from bringing Mr. DeSantis into the fold: It would make it harder for him to attack Mr. Vance from the right.

That debate was defused when Mr. Trump moved on from the notion of bringing on Mr. DeSantis. But the early speculation about the future of Mr. Trump’s Make America Great Again movement — and, specifically, of Mr. Vance’s potential to be Mr. Trump’s political heir — is not limited to those around the vice president.