The striker was forced off in the second half on Tuesday following a challenge from Fabian Delph, but Son Heung-min later scored to give Spurs a 1-0 victory in the first leg at Tottenham Hotspur Stadium.
A club statement read: "Investigations are ongoing and he will continue to be assessed by our medical staff over the coming week."
Spurs also confirmed that Dele Alli fractured his left hand during the game before being substituted in the closing stages.
Tottenham face a fierce battle for the top four in the Premier League, sitting just a point ahead of fifth-placed Arsenal at present, as well as their Champions League campaign in the closing weeks of the season.
It would be a bitter blow for Mauricio Pochettino's side if they were to lose their top scorer for the run-in and the Argentine said he feared just that after the game on Tuesday, although the duration of Kane's absence is still to be confirmed.
"We need to check tomorrow but it looks like it is the same ankle and similar injury," Pochettino told BT Sport.
"It is very sad and very disappointing. We are going to miss him - maybe for the rest of the season. It is a worry for us. We hope it is not a big issue. But there is not to much time to recover.
"He twisted his ankle so we will see how it reacts in a few hours.
"Fabian Delph was very disappointed but he didn't realise Harry's intention was not to tackle him. In the action, both were very strong.
"But both didn't have the intention to make damage to another. That was why Fabian was trying to talk to him. Both were fighting for the ball."
After Tottenham host Huddersfield on Saturday they face back-to-back encounters away to City in the Champions League and Premier League respectively.