The 21-year-old German international defender will officially become a City player on 1st July having agreed a five-year deal with the club.
"This is a very good signing for Manchester City and I am pleased to bring Jerome to the club," City boss Roberto Mancini has said.
"He is a young player who is already of a very high quality, but I think he can continue to improve at City.
"He can play a number of positions in defence, which is important. He has proved his ability in Germany and I think he can be a very important player for Manchester City in the Premier League.
"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."