Kane sustained the injury to his left ankle in the closing moments of the 1-0 loss at Wembley and scans have now confirmed the extent of the problem.
He will miss vital games, starting with Sunday's clash with Fulham and next Thursday's Carabao Cup semi-final second leg against Chelsea.More news: Robertson: Salah diving claims 'a bit of desperation'
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The club said the English forward "is expected to return to training in early March". Given that schedule, Kane may have been rested for the FA Cup fourth-round match against Crystal Palace.
Son, who's scored eight league goals for Spurs this season, could be away from the English side until the end of January.
England boss Gareth Southgate will hope Kane is back in contention for England's opening Euro 2020 qualifier against the Czech Republic on Mar 22.
His absence will be made all the harsher by the departure of Son Heung-min, who joined his South Korea national team colleagues at the AFC Asian Cup in the United Arab Emirates.