WASHINGTON—Workers at the Consumer Financial Protection Bureau arrived at the office Monday to find two acting directors asserting dueling authority and facing off over the direction of one of the nation’s financial watchdogs.

The acting director appointed by the Trump administration, Mick Mulvaney, arrived a little before 7:30 Monday morning with two aides and a bag of doughnuts to work in the director’s office across the street from the White House complex. The office had been vacated Friday by the departure of former Director...