Mick Mulvaney, the acting White House chief of staff, as recently as late last year explored the possibility of becoming president of the University of South Carolina, four people familiar with the discussions said.

Mr. Mulvaney, a congressman from South Carolina for six years before joining the Trump administration, initiated a discussion with a senior official at the university late last year about the position, which is going to become open this summer.

By then, Mr. Mulvaney already had two other jobs — he led the federal Office of Management and Budget, as well as the Consumer Financial Protection Bureau. But he was weeks away from getting a third job that he had lobbied President Trump for over several months: White House chief of staff.

Mr. Mulvaney got the job in an “acting” capacity — a move Mr. Trump said over the weekend gave him “flexibility” with various appointments — after being replaced with a permanent director at the consumer bureau. But chief of staff is not a cabinet-level position requiring Senate confirmation, so it is unclear why the “acting” designation has remained.