Kane was withdrawn inside an hour in the UEFA Champions League quarter-final first leg having suffered "significant lateral ligament" damage.
Per BBC Sport, Pochettino said:
"There is five weeks [of the season left], it is going to be difficult.
"Next week we are going to see with the specialist. We are going to do everything to recover him as soon as possible, but there is only five weeks of the season to recover.
Kane suffered the injury when he slipped while attempting to prevent a clearance from Fabian Delph.
As Metro's Oliver Young-Myles demonstrated, Kane has had several ankle issues in recent years:
Spurs play Everton on the last day of the Premier League season on May 12, so it seems unlikely Kane will play again this season unless the Lilywhites reach the Champions League final, which will played on June 1.
The England captain will then be needed on international duty for the UEFA Nations League semi-final against the Netherlands on June 6.
Losing Kane could be a significant blow to Tottenham, given he's scored 24 goals this season.
They won each of the four league matches he missed through injury earlier in the campaign, though, so if the likes of Heung-Min Son can be at their best again in the coming weeks, they can manage somewhat without him.