Kane enjoyed a fantastic breakthrough season in 2014-15, scoring 30 in all competitions, 21 in the Premier League, making his England debut and being voted the PFA Young Player of the Year.
Sky sources understand Manchester United have an interest in Kane, although no bid has been made while the 21-year-old told Sky Sports News HQ that he was flattered by their interest but is happy at White Hart Lane
But Sheringham, who scored 98 league goals for Tottenham, said staying at the club would be the right move for Kane, comparing him with former Spurs team-mate Jurgen Klinsmann.
Speaking to Sky Sports News HQ, Sheringham said: “Harry reminds me a little bit of Jurgen Klinsmann to be honest.
“When Jurgen came to the club he was full of enthusiasm, Harry’s got that in abundance. He’s a raw talent that is being helped along by his Tottenham colleagues as well.
“He was outstanding last year, it’s a tall order for him to repeat it again this year. He’ll be thinking about going one better, but if he repeats exactly what he did last year then it will be a fantastic season.
He added: “Whenever the question has come up about him leaving he has always said he is quite happy at Tottenham.
“It would be the right move for him to stay at Tottenham and have another steady season and keep doing what he was doing last year; that would be perfect for his development as a footballer.”