Kicker says the Japan international, who has rejected a new contract offer from Dortmund, hopped onto a Manchester-bound plane last night.
Kagawa has already declared he expects to sign for Manchester United.
Dortmund coach Juergen Klopp admits they were always fighting a losing battle: "You have to understand Shinji's environment. In Japan we don't have a profile. It's all about the Premier League and therefore that's where Shinji wants to play."