The England captain admitted over the weekend that he wouldn’t stay at Tottenham ‘for the sake of it’, sparking widespread speculation that he could leave the club.
But Wright, who won four major honours with north London rivals Arsenal – including the Premier League and FA Cup double in 1997-98 – believes Kane should give Spurs one more year to quench his appetite for silverware.
"Now they have a manager [in Jose Mourinho] who is a notorious winner of trophies," Wright told BBC Radio 5 Live.
"Kane did say 'next year will be the first chance the gaffer will have to bed his values into the team'.
"That says to me he is somebody who is already thinking of next year, so he's not going anywhere.
"I'd give this manager a chance with his pre-season, do next season and see where it goes. Because from when he made his debut up to now, I think he has given them everything he can."
But then what? Spurs last won silverware in 2008 with their League Cup triumph over Chelsea, and Wright feels that should that run continue Kane will have to consider leaving his boyhood club.
"If nothing happens and Harry Kane decides to leave then you have you have to say 'well, he's under Mourinho, nothing has changed'. He wants to win medals," added Wright.
"You can't have that kind of ability, that kind of goalscorer who is able to win matches, and not win anything. You can't."