Gallas, who made 78 appearances for the Lilywhites, believes time is now running out for Kane to win trophies and has urged the club to sell their talisman to a "bigger club".
"I hope it happens for Harry next season. I trained with him at Tottenham when he was very young. Even as a young player, he knew what he wanted. He knew he wanted to be one of the top strikers in the Premier League. He has already won a few golden boots," the Frenchman told Genting Casino.
"When [Jose Mourinho] was there Harry was different. He thought that Mourinho could give him the opportunity to win something at the end of the season. Mourinho has a winner's mentality and that matched very well with Harry.
"You need to have the winner's mentality. You need to have the desire to win. Now he has to finish his season with Tottenham and next season, yes, he has to leave, and I hope for him he's going to leave.
"I said in the summer he should leave Tottenham and to go to a bigger club that can give him the opportunity to lift a trophy in this career."
Kane, who has scored 224 goals in 343 appearances across all competitions for Spurs, is yet to win a major trophy in his career since making his senior debut for the Lilywhites in 2011. body check tags ::