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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