Kane collided with opposition goalkeeper Asmir Begovic when prodding the ball into the back of the net, only for the offside flag to deny him a goal.
After initially staying down in some discomfort, the Premier League's joint-top scorer made his way down the tunnel and is now seen as a fitness doubt for club and country.
Spurs have one more game to play - away at Swansea City in the FA Cup quarter-finals next weekend - before the two-week international break.
England are due to take on the Netherlands in Amsterdam on March 23 and Italy at Wembley Stadium four days later.
Kane injured the same ankle twice in the 2016-17 campaign, missing seven games on the first occasion and a further month later in the season.