Though personal terms have not yet been agreed with the Japan international - who has expressed a desire to move to England - a transfer does appear to be moving ever closer.
Reports in Germany on Tuesday suggest Dortmund have now accepted United's offer for a player who refused to sign a new contract after winning back-to-back league titles with the German club.
Dortmund have also expressed a willingness to finalise the transfer.
"Both sides want to conclude the deal," Dortmund chief executive Hans-Joachim Watzke told Der Westen.
Kagawa cost Dortmund only 350,000 euros when joining the club from Cerezo Osaka in the summer of 2010.
This season the 23-year-old scored 16 goals and claimed 10 assists in all competitions for the German double winners.