Tuchel, who won the Champions League with Chelsea and has also coached Paris St Germain, will become Gareth Southgate's permanent successor on January 1, with England skipper Kane having also played under him at Bayern Munich last season, when he scored 44 goals in 45 games..
England's record goalscorer wrote on X: "Looking forward to playing under the boss again!"
Tuchel will be England's third foreign coach after Sven-Goran Eriksson and Fabio Capello.
But former England defender Gary Neville, who rates Tuchel highly, fears his appointment could have a damaging impact on coaching in the country.
Neville told Sky Sports News: "He is a great coach, he's got a proven track record. He's a winner and he knows he can work with big players and handle big personalities. He ticks every single box.
"I'd love Thomas Tuchel to be a coach in the Premier League. We need the best coaches, we thrive on that. You can't deny what Thomas Tuchel is.
"All I'm saying is English coaching is in a worse position today than it was yesterday.
"I think everyone will wish him all the best and hope he can get over the line and win a trophy but I think there are some serious questions for the FA to answer in respect of English coaches. I do think we are damaging ourselves.
"With the likes of Graham Potter and Eddie Howe, I do think there are outstanding candidates that could have been appointed that are English."
Alan Shearer believes the FA acted quickly to bring in Tuchel because of the possibility he could become Manchester United manager.
Tuchel's nationality does not sit well with some fans and another former England forward, Gary Lineker, admits he has mixed feelings over that issue.
Lineker said: "I don't think it is imperative (to have an English manager). I'm not going to lose sleep over it but my personal preference would be that England has an English coach.
"For some reason English football hasn't produced a plethora of brilliant coaches. Why that is, I don't know.
"I respect Thomas Tuchel as a coach. I think he's really clever. No-one will care one iota (about his nationality) if England somehow manage to win the World Cup.
"This is an unbelievable opportunity of winning the greatest trophy of all and boy would I love him to do it."
Another former England striker, Michael Owen, met Tuchel's appointment with enthusiasm, writing on X: "Top quality appointment made with the new @England manager.
"A proven winner with a great understanding of the English game. My only disappointment is that he's not English."
Meanwhile Prince William, who is patron of the FA, has welcomed Tuchel to the job.
The Prince of Wales wrote on X: "Exciting times for @England, with a generation of talented players and a new manager taking the reins. Thomas, wishing you the best of luck, we're all behind you! W"
