The German champions announced on Sunday that Saibari had undergone surgery after sustaining an injury to his right ankle during the Klassiker.
The midfielder was forced off during the match, which ended in a draw at the Allianz Arena. Bayern Munich subsequently confirmed that the 25-year-old had suffered a syndesmosis ligament tear and undergone an operation.
Gnabry also left the game with an injury, with the club confirming that the Germany international had sustained a muscle injury in the back of his left thigh.
The two setbacks are a further blow for Bayern, who were already managing a number of absences in their attacking options.
The club have not provided a definitive return date for either player, meaning their availability for the demanding run of fixtures ahead remains uncertain.
Vincent Kompany's side must now assess their options as they prepare for Tuesday's Champions League meeting with Atlético Madrid. The injuries also add to the questions surrounding Bayern's squad depth, particularly with the season entering a busy period.