Turkish newspaper Hurriyet - as reported by Le 10 Sport - claims Galatasaray are plotting a permanent deal for the out-of-favour United midfielder.
The Japan international has struggled for first-team football under Old Trafford boss David Moyes, and was linked with a move to former-club Borussia Dortmund earlier this season.
Jurgen Klopp's team are seen as the most likely destination should Kagawa opt to leave United, although publicly Kagawa insists he's resolved to stay and fight for his place at Old Trafford.
He's admitted he has not performed at his expected level since joining the club, but wants the opportunity to stake his claim for a first-team place under Moyes.
Still, the report claims Galatasaray are willing to give him an escape route, with new boss Roberto Mancini expected to be active in the January window.
Current Gala star Wesley Sneijder has been linked with a move to the Premier League, and Kagawa could be the man to replace him in Istanbul.
With United short in midfield though, it remains to be seen whether Moyes would be willing to let Kagawa go.
The Japanese star may be struggling for form, but there's a feeling it's only a matter of time before he rediscovers his Dortmund form at Old Trafford.