Striker Kane sustained a significant ankle ligament injury against Manchester City last week, a similar knock to the one which kept him out for six weeks earlier this season, but Spurs confirmed on Thursday that he has begun his rehabilitation.
Pochettino would not put a timeframe on his likely return, only to say the England captain will do everything possible to make it back before the end of the season.
“Still we cannot say nothing,” he said. “We are going to assess him day-by-day, we’ll see.
“We’re not optimistic but you know Harry Kane and with Harry Kane all is possible.
“We’re not going to say he won’t play again or not, we are going to assess him day-by-day.”
Of course, Kane has a massive incentive to get back as quickly as possible after Spurs qualified for the Champions League semi-final for the first time in their history with a dramatic away-goals win at City on Wednesday.
After losing 4-3 on the night, they progressed with a 4-4 aggregate score on a memorable evening in Manchester.
Spurs were seconds from going through, then thought they had been dumped out when Raheem Sterling appeared to have scored in stoppage-time, only to enjoy a euphoric celebration when VAR ruled Sergio Aguero was offside in the build-up to the goal.
The turnaround in emotions was an unbelievable sight as Tottenham players that were on the floor crestfallen were suddenly punching the air, while City, who celebrated as wildly as they did when Aguero won them the Premier League in 2012, were in despair.
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Man City post premature Raheem Sterling goal celebration tweet and fans reacted brilliantly
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Harry Kane posts hilarious celebration video
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Video: Spurs celebrate in remarkable scenes following thrilling Champions League
Pochettino also revealed how he went from ‘thinking very bad things’ to being ‘alive again’.
“I didn’t want to watch because in that moment I take off my jacket and jumper and throw to bench and go to my seat and think very bad things,” he said.
“But in that moment I hear Jesus’ (Perez assistant manager) voice say, ‘Oh maybe it is offside, VAR is checking’, and in that moment I was alive again.
“I understand the situation on the City bench; from top of the world down, disappointed, that is unbelievable, the emotion we live last night was amazing.
“A very good end for us but I feel sorry when that situation happens because you can be on the other side and it is difficult when that situation happens.
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“Last night it was for us, all you can do is enjoy, celebrate, be respectful of the opponent.”
Spurs do not get long to bask in their glory at the Etihad Stadium as they return there on Saturday lunchtime for a Premier League clash.
Pochettino admits his toughest task is to get his players to focus on the game, which could have big implications on both City’s title ambitions and Spurs’ top-four hopes.
“You know that we left yesterday everyone is happy but a little bit tired too,” he added.
“We arrived (home) late, we didn’t sleep too much but the happiness allows you to feel strong and with energy, and of course now we’re thinking about the game on Saturday.