The England captain, who was out for a more than a month after sustaining ligament damage in January, rolled his ankle and was then caught on it by Fabian Delph in the quarter-final first leg.
Spurs went on to win the game 1-0 courtesy of a second-half goal from Son Heung-min.
Spurs boss Pochettino said: “If you see the action it is an action that can damage the ligament again. We need to check in the next few days.
“It is so painful. We hope it is not a big issue and try to recover as soon as possible but it doesn’t look good.”
Asked if he was worried Kane’s season could be over, Pochettino said: “Yes.”
Kane left the stadium using crutches and with his foot in a protective boot. Pochettino spoke to Delph after the incident, which occurred just before the hour.
Asked about that conversation, Pochettino said: “I tried to make him understand that it wasn’t the intention from Harry or him to damage each other. It was a tough action.
“I said to him, you were lucky - checking on the VAR, maybe technically it was a red card - but I think both were fighting for a ball and no intention to damage each other.”
Spurs now hold the advantage ahead of next week’s second leg at the Etihad Stadium but Pochettino, despite the deserved victory, still regards City as the favourites.
He said: “Of course I am so happy. I am happy with the performance and the way we approached the game.
“We still have a lot of work to do. Manchester City are still the favourites to win the Champions League and it is going to be tough.”
Elsewhere, midfielder Naby Keita looks like he is finally starting to deliver on his £53m price tag after scoring for the second successive match to put Liverpool in control of their quarter-final with Porto.
The Guinea international netted his first Anfield goal in the 2-0 win over the Portuguese visitors five days after his maiden strike in the Premier League at Southampton.
Keita put in arguably his best performance in a red shirt since joining in the summer and after a difficult settling-in period appears to be fulfilling the faith Jurgen Klopp placed in him in agreeing a deal with RB Leipzig 12 months before he actually arrived.
Roberto Firmino’s 14th goal in the competition for the club, drawing level with Mohamed Salah and Ian Rush, doubled Liverpool’s lead but the only disappointment on the night was that they did not finish the job as they had done last season when they won the first leg 5-0 in Porto.