The warning has come from Dimitar Berbatov who quit north London for Old Trafford in 2008.
Berbatov clinched two Premier League titles in four years at United and believes Kane will satisfy his lust for silverware by making the same move.
“I see a lot of similarities in the situation I went through and what is happening with Kane,” said the Bulgarian, who moved to Manchester for £30.75million.
“I feel that it’s also a similar time in his career that I was at, you start to think if you’re on the right path and you think about winning things.”
United are among a number of clubs with a long-standing interest in Kane whose season was curtailed with a torn hamstring in January.
Tottenham ’s England captain had scored 27 goals in 31 appearances for club and country before his injury. Without him, Spurs crashed out of the FA Cup, the Champions League and were struggling for form and consistency in the race for a top-four place.
Kane has four years left to run on the £200,000-a-week deal he signed in 2018 but that has not prevented some of the biggest clubs in Europe being interested.
Nor did it stop the Chingford-born marksman from admitting for the first time last week that he could leave if his club don’t start winning trophies.
Kane said: “I love Spurs, I’d always love Spurs, but it’s one of those things. I’ve always said if I don’t feel progression in the team or it’s going the right direction then I’m not someone to just stay there for the sake of it.
“I’m an ambitious player, I want to improve, I want to get better. I want to become one of the top top players.
“It all depends what happens with the team and how we progress as a team. So it’s not a definite I’m going to stay there forever, but it’s not a no either.”
Berbatov recognises the dilemma facing Kane with Spurs having made him a Premier League superstar during his two years at the club before his defection.
As well as winning two titles at United, Berbatov reached the Champions League Final in 2009, losing to an outstanding Barcelona team. He also clinched the 2010-11 Golden Boot.
The Bulgarian told Betfair: “When I left Spurs, even though I had won a trophy with the club it was a still difficult decision to make.
“I was following my own path and I knew that people wouldn’t be happy, but I had to do it, I come from a small country and when one of the biggest clubs in the world came in for me, it was my dream.
“There is a strong connection between Spurs and Manchester United, Teddy Sheringham was at Tottenham then went to Old Trafford, I was a Spurs striker and went to United and now Kane, who’s also a striker, is being linked with a move there.”