The Tottenham striker came on in the 79th minute of England's 3-1 victory in the UEFA Under-21 European Championships qualifier, but was withdrawn just six minutes later after suffering the injury.
Manager Gareth Southgate had planned to take Kane to the Toulon Tournament, which starts later this month, so the former Middlesbrough boss was left to rue the 20-year-old's absence.
"It's a real shame because Harry has been extremely patient for his chance and I was looking forward giving him some of the prestigious games in Toulon," he told the Football Association's official website.
"He's warranted that opportunity and has had a super finish to the season with Tottenham so it's disappointing for us and for him."
Kane scored three goals in six Premier League starts for the London club this season.
