Interim head coach Hansi Flick confirmed the winger could be sidelined for upwards of four weeks after metal screws were inserted to fix the damaged joint.
According to Bild, Perisic was taken away for treatment after Odriozola, who joined Bayern on loan from Real Madrid last month, accidentally hurt his right ankle during a training session early on Tuesday.
Speaking to reporters ahead of Wednesday's DFB-Pokal third-round clash with Hoffenheim, Flick said: "We thought it wasn't that bad, but he has a fracture in his outer ankle.
"It needs to be screwed back on, which means he is out for four weeks and only then can he start building up training again.
"These things happen in football, but we'd have liked it to have been different."
The news is a setback for Bayern, who have won six Bundesliga games in a row to return to the top of the table.
The champions face a difficult week, with the visit of Hoffenheim followed by a clash with title rivals RB Leipzig on Sunday.
Fortunately for Flick, forward Serge Gnabry is "100 per cent fit" once more, while defender Lucas Hernandez is close to a return and Kingsley Coman could be back for the Leipzig match.
"He will be in the squad tomorrow," Flick said of Hernandez. "He is making progress, but it's important for him to be there, so he can get some game time."
Bayern's form prompted president Herbert Hainer to suggest Flick could be given the job on a permanent basis at the end of the season, telling Passauer Neue Presse: "If he keeps winning like this, there is no alternative."
Asked about those comments, Flick said: "I know Bayern can be different from week to week. Obviously, you're happy to read something like that.
"Bayern have to see for themselves what they want for the future."
For now, Flick is focused on reaching the next round of the DFB-Pokal, a challenge he hopes will be made easier by the recent form of Thiago Alcantara, who has scored in each of Bayern's past three league games.
"He's an outstanding footballer, he can play both as a six and an eight and he has his attacking qualities," Flick said.
"He said before the season that he can score eight goals, so he has five more to go!"