A Bola reports that Kagawa could return to Borussia Dortmund in January in a swap deal which will see 24-year-old defender Neven Subotic join United.
The Japanese international has failed to make the expected impact at Old Trafford under new manager David Moyes.
Borussia Dortmund boss Jurgen Klopp is keen for Kagawa to return to Germany, where the Japanese playmaker was a key player.
Moyes could risk the wrath of United fans, should he decide to sell Kagawa as they are eager for him to become a first-team regular.
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Liverpool have unveiled their new home kit for next season, sticking with the hugely popular retro styling of the current shirt.
The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.