The Argentine, who was sacked at Spurs in November 2019, will begin his new role at Stamford Bridge on July 1st after signing a three-year contract in west London.
Such is the rivalry between the two clubs, Pochettino has invoked the fire of Spurs fans for his decision to move to Chelsea.
Kane, who shot to prominence, under the former Southampton manager was speaking at a Reuters event in New York.
“Mauricio was an amazing manager for me,” he said.
“Great person, great, great coach. He helped me a lot to get to where I am now. So I’m really appreciative of him.
“Everyone has their career, everyone has their future. So all I can say is I wish him all the best.
“I hope he does well – just obviously not as well as us!”
A hugely popular figure at Tottenham during his spell in charge, there had been calls from supporters to bring the 51-year-old back to the club.
Spurs, however, will instead accelerate their pursuit of Celtic manager Ange Postecoglou after the Scottish Cup final this weekend.
The 57-year-old is said to have impressed Daniel Levy after restoring Celtic’s dominance in the Scottish Premiership, winning two League titles in his two years with the Glasgow giants.