Spurs gave no timeframe for when Kane might be fit again.
"Following assessment over the past two days, we can confirm that Harry Kane has sustained a significant lateral ligament injury to his left ankle following a challenge in our Champions League quarter-final, first leg win against Manchester City," Spurs said on their website.
"Investigations are ongoing and he will continue to be assessed by our medical staff over the coming week."
The England captain, who has scored 24 goals in all competitions this season, left the club's new stadium on crutches with a protective boot on his foot.
On Wednesday, 25-year-old Kane had tweeted: "Gutted to go off injured but every setback is a chance to come back stronger than ever. Big finish from the boys to go on and win!"
Spurs also said England midfielder Dele Alli had fractured his left hand in Tuesday's game and was being assessed to see if he can play against relegated Huddersfield Town on Saturday.
Fourth-placed Tottenham are battling for a top-four finish to claim a Champions League spot for next season. They trail third-placed Chelsea by two points, with a game in hand, and lead fellow London rivals Arsenal by one point.
(Reporting by Shrivathsa Sridhar in Bengaluru and Ken Ferris in London; Editing by Toby Davis)
