Joe Lonsdale, a tech investor close to Mr. Musk, is not planning to play a formal role in the Trump transition, according to a person briefed on the matter, and he has said publicly that he does not want to join the administration “full time” but would rather play a part-time advisory role.

Musk has spent days bounding in and out of meetings, sitting in on phone calls, weighing resumes and traveling to Washington as Trump’s unofficial co-president.