Kane was originally forecast to be sidelined until early March after suffering ankle ligament damage against Manchester United last month but the England captain has always been confident of making a quick recovery.
After spending a week in warm weather training in Barbados, Kane returned to Spurs on Tuesday to begin ball-work.
The Spurs manager played down Kane's chances of being fit for Sunday's visit of Leicester or the Champions League last 16 first leg against Borussia Dortmund on Wednesday but said he could "maybe" play at Turf Moor.
"Now he is here, he is doing well, he is doing fantastic," Pochettino said on Friday.
"We need to stop him every day because he wants to be ahead of his recovery and he’s very optimistic and determined to play as soon as possible.
"It’s unbelievable but he’s doing well, if you see him he’s nearly ready to play but we need to be realistic too and of course in between all the areas to take the best decision. He’s so close to come back again. Not for Sunday, not for Wednesday but we will see after. Maybe [for Burnley], maybe yes."
Spurs also said they expect Dele Alli to return from a hamstring injury in early March and Pochettino played down the England midfielder's chances of returning ahead of schedule, after he too started running and ball-work in Dubai this week.
“It's a different injury [to Kane's] and I think we need to look at it in a different way," Pochettino said. "He is doing well and we are happy with the way that he is recovering."
Meanwhile, Pochettino has hit out at the Premier League for leaving Spurs at a "massive disadvantage" for next week's Champions League last 16 first leg against Borussia Dortmund.
After a week off, Spurs face the Foxes on Sunday, while Dortmund host Hoffenheim on Saturday, leaving them with 24 hours more rest.
"I'm a little disappointed with the Premier League because we are not receiving help," he said. "Because it is clear that it is a massive disadvantage to play with one day less to prepare.
"When you have a clean week, why not play today or play Saturday when you are going to play a decisive game against Dortmund? When you play Sunday and then it is Monday and Tuesday can’t do too much as players involved on Sunday need to recover. That is massive. It is difficult to understand. I don’t like to complain but of course the Premier League must care about teams involved in a very important competition like the Champions League."
Asked if the club had spoken to the Premier League, Pochettino said: "No. Because I am not in charge and I don’t know how there a way. But I think when the TV decide the game I think you cant move or charge.
"In another country – Spain Germany or France – clubs that are involved in European competition always they are helped. The league is going to help you to arrive in the best competition.
"When you play you need minimum 72 hours if you want to start to recover in a very good way to compete but we are going to have less."