Davies was challenging Dortmund midfielder Jude Bellingham for the ball towards the end of the first half and his head came into contact with the English midfielder's raised boot.
Davies, a Canada international and a key member of the national team that qualified for the World Cup for the first time since 1986, was withdrawn at halftime.
Bayern did not give a timeline on his return to action but said he was "doing well under the circumstances" and sat out training on Sunday.
The 2-2 draw left both teams on 16 points with Bayern third and Dortmund fourth in the standings.