The former Tottenham striker says Kane will thrive on the bigger stage at White Hart Lane next season.
Kane proved that his superb breakthrough campaign at Spurs in 2014-15 was no flash in the pan with an arguably even more successful season this term.
The 22-year-old became the first Englishman in 16 years to win the Premier League golden boot with 25 goals from 38 matches, and has established himself as the key man for the England national side ahead of Euro 2016.
The striker will face a new challenge next season as he looks ahead to playing in the Champions League for the first time in what will be his third full season spearheading the club's attack.
Klinsmann, who scored 39 goals in 68 matches for the North London side, believes that challenge, along with starring for England at Euro 2016, will prove a huge boost for his development.
The former Germany striker said, as quoted by the Sun: "I adore Harry Kane. I love him. I really like the way he has developed over the last few years.
"To have Champions League football for Harry is very important for his development as a player. That is a huge challenge as is playing in the Euros this summer.
"It’s a big leap forward but it’s fantastic for him to have the opportunity to show what he can do."
Kane will be hoping to showcase his ability and transformation into a top European striker by leading England's charge at Euro 2016 later this month.