Kane limped off with four minutes remaining in Bayern's 1-1 home draw with Bayer Leverkusen on Saturday.
The 31-year-old, who has a history of ankle injuries, was seen clutching at his lower leg after a collision with Leverkusen's Amine Adli.
In a statement, Bayern said Kane "suffered a painful blow to his ankle", but added "examinations by (the club's) medical department on Sunday showed a positive development".
"Kane will continue to receive intensive treatment."
Bayern sporting director Max Eberl said Saturday: "When Harry leaves the field it means something because he's pretty tough.
"(The incident) doesn't look good, but we hope he's made out of good English wood and nothing's broken."
Kane has been in great form for Bayern this season.
He has scored 10 goals in seven games in all competitions, including hitting four in Bayern's Champions League opener against Dinamo Zagreb.
