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 quarterfinal, 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."
Kane tweeted on Wednesday that he was "gutted to go off injured but every setback is a chance to come back stronger than ever."
Following Tuesday's win, Spurs boss Maurio Pochettino said: "It's very, very sad. It's going to be tough and we're going to miss him -- maybe for the rest of the season."
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. Kane had previously missed six weeks of action after injuring the same ankle during a Jan. 15 Premier League match against Manchester United.
The club also confirmed that Dele Alli fractured his left hand during Tuesday's fixture.
The England midfielder will be assessed to determine his availability for this weekend's match against Huddersfield Town.
Information from Reuters was used in this report.