Supporters had feared the worst after losing the striker for six weeks earlier this season to a similar problem, but the club does not believe this setback to be as severe as his previous one.
Spurs tweeted: "We can confirm that @HKane sustained ligament damage to his right ankle during Sunday's FA Cup win against Millwall.
"The injury is similar to the one suffered by the forward against Sunderland in Sept last year, however it is not considered to be as severe.
"Harry will continue to be assessed by our medical staff while he undergoes his rehabilitation at our Training Centre."
Kane is likely to miss England's friendly in Germany on March 22 and a home World Cup qualifier against Lithuania four days later.