Kane, who has scored 24 goals in 29 appearances this season, suffered the problem during his side's FA Cup victory over Millwall earlier this month.
Early reports suggested the injury was set to sideline Kane for the remainder of the campaign but Standard Sport understands the England international is hopeful he will be able to start April's FA Cup semi-final against Chelsea.
The injury is not as severe as the one that kept Kane out for 10 matches in six weeks earlier this season and Pochettino is upbeat about the striker's situation, despite not being able to offer a definitive timeframe in regards to the 23-year-old's return.
“We don’t know yet. He is doing well, he started to work on the pitch a few days ago," Pochettino told his pre-Burnley press conference.
"We are very positive, he is very positive but we’ll see when he’s available again. I cannot provide the time that he still needs. It’s not too long.”
Saturday's trip to Burnley will certainly come too soon for Kane, which means Pochettino must find a deputy to lead his side's front line at Turf Moor.
Vincent Janssen and Heung-min Son are the most likely candidates but the Dutchman has been suffering from illness and missed his nation's midweek friendly against Italy as a result.
However, Pochettino has confirmed Janssen will be available for selection against Sean Dyche's side while Son is also set to be in contention for a starting berth after returning to north London from South Korea.
The Argentine coach said: “Today [Thursday] he [Janssen] was with the group. He is okay after the illness he got with the national team.
"Tomorrow [Friday] we need to assess him but he will be available to be in the squad for Burnley.
“Yes, he [Son] is here. All the players arrived yesterday [Wednesday] afternoon.
"He arrived, different players like Hugo [Lloris], Kevin Wimmer too and today we are all together. We met all together on the training ground this morning.”
Meanwhile, Danny Rose's potential return date remains unclear. The full-back has been sidelined with a medial ligament injury since January and is yet to resume full training.