Kane has graduated from Spurs’ youth academy, to become a first-team star.
However, Carragher feels that Kane’s reputation is growing so rapidly that he will soon become a target for Europe’s biggest clubs.
“Top clubs will start looking at this lad if he has a great Euro’s,” the former Liverpool defender said.
“I think he should stay at Tottenham, because I think under the manager they have got they could go into great things. But I don’t think he will finish his career at Tottenham. He’ll be going to a team which can regularly win Champions Leagues.”
Kane has joined an elite group of strikers who have netted 20 goals in consecutive Premier League seasons.
The 22-year-old is now widely regarded as England’s brightest striking talent, and he has been crucial to Tottenham’s success this season.
Kane has netted 25 league goals, as Spurs have qualified for the Champions League in fine style.
Kane is now expected to play a key role for England at Euro 2016 in France.
Kane has already been linked with the likes of Manchester United and Real Madrid, but it would be a major surprise if he left White Hart Lane this summer.