Davies was substituted after three minutes of Sunday's (AEDT) 5-0 Bundesliga victory at Allianz Arena after rolling his ankle.
Head coach Hansi Flick said the 19-year-old has torn ligaments and is not expected to return until at least early December.
"He sustained a ligament injury. One ligament is fully torn, another is partially torn. We reckon he'll be sidelined for six to eight weeks," Flick said.
"It hurts when a player of this quality is ruled out."
Davies is likely to miss most of Bayern's remaining matches in 2020, including their five remaining UEFA Champions League group games and Der Klassiker with Borussia Dortmund on 8 November (AEDT).
The champion's win over Frankfurt, in which Robert Lewandowski scored a hat-trick before further goals from Leroy Sane and Jamal Musiala, moved it to within a point of league leader RB Leipzig after five games.