Kane left the stadium on crutches and wearing a protective boot after coming off in the second half, before Son Heung-min scored the only goal in the quarter-final first leg tie.
Pochettino admits that the club aren’t optimistic that the England hitman will be back before the season’s end.
Kane was out for seven weeks earlier this year after injuring the same ankle.
And Pochettino said: “We have to hope it is not a big issue but it is so painful. It doesn’t look good. We will check in the next few days but he could maybe have damaged the ligaments.”
Asked if Kane could miss the rest of the campaign, he added: “Yes – we are very sad, very disappointed and it’s going to be tough for the rest of the season.”
The incident occurred after Kane’s tussle with his Three Lions teammate Fabian Delph on the touchline.
Kane was attempting to block Delph’s pass back to goalkeeper Ederson, but off balance appeared to twist his ankle – down onto which came Delph.
The City midfielder-cum-leftback wasn’t pleased with Kane’s attempted challenge, reacting with a degree of anger.
That ultimately led to words with Pochettino, who revealed: “He was so aggressive with Harry after the action, I tried to make him understand that it wasn’t an intention to damage each other, it was just a tough action.
“[And] he stepped on his ankle. I said to him that he was lucky, because checking on VAR maybe technically it was a red card.
“Both were fighting for the ball, the intention was not to damage each other. [There’s] no problem between us.”
Spurs take a 1-0 lead to the Etihad when the pair resume battle in next week’s second leg.