The 21-year-old has signed a five-year contract with City and will officially join the club at the start of July.
“This is a very good signing for Manchester City and I am pleased to bring Jerome to the club,” City boss Roberto Mancini told the club’s official website.
“He is a young player who is already of a very high quality, but I think he can continue to improve at City.
“It’s good for Jerome to have agreed terms ahead of the World Cup, so he can now concentrate on playing for Germany.
“And I will be very interested to see how he gets on in the tournament, before welcoming him back to meet his team-mates in pre-season.”