The 29-year-old centre-back came on for Mats Hummels after 18 minutes but was taken off at half-time in the 2-1 loss, having suffered the problem without the involvement of another player.
Sule will miss this week's German Cup first-round tie at Essen and the first part of the Bundesliga season, with Dortmund kicking off their league campaign at St Pauli on August 23.
The Ruhr valley club, who finished fourth last season, will also compete in the Champions League league phase.