The England captain suffered the injury during the final stages of Spurs' 1-0 defeat to Manchester United on Sunday, with tests having taken place once the injury settled down.
A statement on the club's website read: "Following preliminary assessments, we can confirm that Harry Kane has damaged ligaments in his left ankle, sustained during Sunday's match against Manchester United.
"The striker will continue to be monitored by our medical staff as he commences rehabilitation and is expected to return to training in early March."
The England captain was injured after being caught in the middle of Phil Jones and Victor Lindelof as Spurs chased an equaliser against United.
It is the third successive season in which Kane has suffered an ankle injury, the most recent one keeping him out for a fortnight last March.
The injury is a huge blow to third-placed Tottenham's titles hopes, with the loss to United leaving them nine points behind leaders Liverpool.
Kane is now set to miss the second leg of the Carabao Cup semi-final against Chelsea, which Spurs lead 1-0, as well as at least six Premier League matches, including the trip to Stamford Bridge on February 27, and the first leg of their Champions League last-16 tie at home to Borussia Dortmund.
The striker is also highly likely to be unavailable for the north London derby against Arsenal on March 2 and the return meeting with Dortmund three days later, but could be back in contention for England's opening Euro 2020 qualifier against the Czech Republic on March 22.
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