Kagawa has failed to hold down a regular first-team spot at Old Trafford since sealing a move to the Premier League in 2012.
The 25-year-old attacking midfielder has had to compete with the likes of Wayne Rooney, Ashley Young and Adnan Januzaj for a place in Manchester United's starting line-up.
Kagawa has started Manchester United's last four games - but is still far from one of the first names on his side's teamsheet.
A move back to Dortmund would see Kagawa link back up with German tactician Klopp - who is a big admirer of his former star player.
Klopp revealed last year that he was saddened to see Kagawa play on the left wing rather than in the hole behind a lone striker.
The 25-year-old, who is expected to play a major role for Japan in this summer's World Cup, will find it hard to feature regularly in his preferred position as Rooney is often deployed behind the main striker.
Dortmund are also keen on signing Hoffenheim’s Roberto Firmino and Torino’s Ciro Immobile.