The People reports that Kagawa is keen to leave as he does not want to risk a lack of fitness and form ahead of the World Cup.
The Japanese midfielder has not been a regular at Old Trafford since David Moyes took over as manager.
The 24-year-old has reportedly told team-mates that he needs to leave United in January.
Kagawa joined United from Bundesliga giants Borussia Dortmund in a £17million deal in June 2012.
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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.