The England international was replaced after seven minutes of Sunday’s 6-0 FA Cup quarter-final win over Millwall.
He was hurt when defender Jake Cooper blocked his shot close to the byeline.
Spurs said the injury is similar to the one Kane picked up against Sunderland on 18 September.
The 23-year-old missed five Premier League games and two EFL Cup matches after twisting his ankle tackling Sunderland’s Papy Djilobodji.
Kane is likely to miss England’s friendly in Germany on 22 March and a home World Cup qualifier against Lithuania four days later.
It is not clear if the top flight’s joint leading scorer with 19 goals will be available for Tottenham’s FA Cup semi-final against Chelsea at Wembley on the weekend 22-23 April.