Kane limped off in stoppage time after his two goals earned Spurs a point at Goodison Park and boss Jose Mourinho was unable to give much detail on the severity of the issue after the game.
Mourinho said that a diagnosis would take place this weekend, with a scan likely depending on how Kane feels on Saturday morning.
It is a process that Spurs and Kane know only too well as this is his seventh separate ankle injury since 2016.
The injury puts doubts over Kane’s availability for next Sunday’s Carabao Cup final against Manchester City in what would be a bitter blow for Spurs.
The England captain, who made no reference to the injury in social media posts after the game, again was key for Spurs on Merseyside.
His two goals, which take him clear in the race for the Premier League Golden Boot with 21, masked another dismal performance by Mourinho’s side whose Champions League qualification hopes are fading fast.
They went ahead in the 27th minute with a clinical finish by Kane before Gylfi Sigurdsson’s strikes either side of half-time put Everton in front, only for Kane to level with 20 minutes left.
Sigurdsson’s opener was from a penalty awarded by referee Michael Oliver for a foul on James Rodriguez, who appeared to kick the ground.
VAR declined to overturn the decision, meaning it was the seventh penalty given against a Mourinho team in 22 Premier League games refereed by Oliver.
“I think not to protest should be something to praise and not something to consider a negative thing,” Mourinho said in relation to his players not complaining about Oliver’s decision.
“At least in what we have in every meeting with the football authorities, before matches when captain and one coach, they say don’t surround the referee. Leave the referee alone, let him make the decision.
“On top of that I think we all start having a feeling of what a VAR is. I think it is very difficult for VAR to go against a referee decision. I believe they don’t like to expose each other.
“I believe the maximum they like to tell the referee ‘come on, come and have a look’ so I don’t think they like to go against.
“So when Mr Oliver went with such a high speed to the penalty spot I don’t think there is any point in campaigning too much.”
The draw does not do much good for Everton’s pursuit of European football either though they had chances to win the game, none better than late on when Josh King’s strike was saved by Hugo Lloris and Richarlison put the rebound over.
Sigurdsson, who enjoyed playing against his former club, told the club’s official website: “We’re satisfied with the performance but disappointed not to take the three points.
“Especially with the chances we had, it feels like we dropped two points.
“In our last home game against Crystal Palace (a 1-1 draw on Easter Monday), I don’t feel we played as well as we did here, even though we created a lot of chances in that game, too.
“I thought the performance, the defending and the pressing was a lot better.
“We did a lot of work during the week – even though we didn’t have a lot of time after the Brighton game (on Monday).”