Kane suffered an injury at the end of Sunday's 1-0 defeat to Manchester United and limped off the Wembley pitch very gingerly, while Spurs are now losing Son to international duty and could be without the South Korean until the competition's final on Feb. 1.
"My worry is that it was a bad tackle in the last moment," Pochettino told the BBC. "It wasn't the intention of the United player but it was a bad tackle and now he has a bit of a swelling on his ankle and he was limping after the game.
"We are going to lose Son for the Asia Cup and if Harry Kane suffers an injury, it is going to be massive for us."
Moussa Sissoko also limped off at the end of the first half with a groin problem as Spurs' injury-hit season continued.
"That is the risk in any competition," said Pochettino in his post-match news conference. "The people, they criticised me when we played Harry Kane for 15 or 20 minutes at Tranmere. I told you the injury can appear when you don't know when, and look today, [it happened] in the last action of the game.
"First of all we need to assess him and Moussa and hope it's not a big issue and then we're going to try and find a way to play if we are to lose them, but we hope it's not going to be a long period. I think Moussa is only a small thing but we will see. It was a small thing in his groin."
Tottenham striker Vincent Janssen has recovered from a foot injury and scored in an Under-23 match against West Ham on Sunday. But, asked whether the Dutchman has a chance of rekindling his Spurs career and making his first senior appearance of the season, Pochettino replied tersely: "No, he's not in my plans."
Pochettino professed himself to be pleased with Spurs' performance against United despite the defeat, having seen his side muster 21 shots, with 11 on target, but fail to beat David De Gea.
"The performance I am so happy with," he said. "The last 45 minutes, the second half, we played so well. I think we created enough chances to score and win the game. De Gea made 11 saves which made him man of the match in the end.
"For me it was one of the best performances that I've seen, after four and a half years, at Tottenham -- the best 45 minutes.
"My feeling today is in the opposite way to when we played at Old Trafford in Manchester. We won that game 3-0 but I wasn't happy. Today, after the defeat, I'm very pleased with the performance.
"If you want to build a team and a structure for the future and be close to winning titles, that's the way you have to work and play.
"I'm disappointed with the result of course -- I cannot talk in a different way -- but I'm very happy with the reaction after we conceded the goal and in the second half we created more than enough chances to score, to draw or win the game."