He limped off during the defeat at Southampton on Wednesday and on Friday manager Jose Mourinho said he was expecting “bad news”.
No timescale has been given for his return but he is likely to miss several weeks of action, including the FA Cup tie at Middlesbrough on Sunday.
“Head up. Tough times don’t last, tough people do,” Kane tweeted.
“Everybody knows who he is, what he is, what he means for the team, the fans, the club,” Mourinho said.
“His quality, the routines that the team has playing with him, every minute of every game he doesn’t play we miss, so every match that he doesn’t play we’re going to miss him.”
In a statement, Spurs said: “We can confirm that Harry Kane suffered a tear in his left hamstring during our New Year’s Day fixture against Southampton.
“Our medical staff will continue to review the injury with treatment ongoing.”
Kane has scored 27 goals in 31 appearances for club and country this season.