WASHINGTON—An appeals court appeared skeptical Thursday that the interim director of the Consumer Financial Protection Bureau could independently run the bureau given his other role as head of the White House’s Office of Management and Budget.

A three-judge panel heard oral arguments in a case challenging the authority of Mick Mulvaney, a Trump appointee, to head the CFPB as acting director. The title of acting director is also claimed by Leandra English, who has worked for the CFPB since the Obama administration. Ms. English...