Kane has established himself as one of the best forwards in the world, with the 24-year-old's form resulting in alleged interest from Real Madrid.
However, Mason - who has recently retired from football due to a fractured skull - predicts that Kane will want to break Alan Shearer's goalscoring record in the Premier League before contemplating an exit from the North London outfit.
The former midfielder told talkSPORT: "What I believe is that Harry wants to go down in the record books as a record-breaker and going out to La Liga that's not going to happen.
"Staying in the Premier League with Tottenham, that will happen. He'll break Shearer's record, he'll break [Wayne] Rooney's record for England – I'm so confident.
"He doesn't need to win a trophy right now. Harry I believe will stay at Spurs, he will stay there scoring goals, and I believe he will win trophies with Spurs."
Kane has scored 134 goals in 204 appearances for Spurs in all competitions.