Swedish prosecutors reopened their investigation into a rape allegation against Julian Assange on Monday, a development that could frustrate the U.S. government’s yearslong effort to bring the WikiLeaks founder to trial over a massive leak of classified documents almost a decade ago.

The decision opens the door to a tug of war between Swedish and U.S. authorities over who gets to put Mr. Assange on trial. He is currently serving 50 weeks in the U.K. for skipping bail in 2012 and hiding out in the Ecuadorean embassy for seven-and-a-half...