The 30-year-old has been the subject of enquiries from the Old Trafford club, according to the Mail Online.
Muller is said to be unhappy with his lack of opportunities under Niko Kovac with the Bundesliga giants this season.
United are long-time admirers of the Germany World Cup winner and previously made an attempt to sign him during Louis van Gaal’s spell as manager.
However, the club’s inability to offer Champions League football could scupper their hopes of signing the player, who is also wanted by Inter Milan.
The Serie A side have registered interest in the player as they look to find back-up for on-loan Manchester United forward Alexis Sanchez, who is injured.
A January move for Muller could prove complicated as Bayern club chairman Karl-Heinz Rummenigge has insisted he is still an important part of the team and the player is under contract until 2021.
Philippe Coutinho’s summer arrival on loan from Barcelona has restricted Muller’s game time, as he has been restricted largely to substitute appearances.
Comments from Bayern president Uli Hoeness are also said to have annoyed the player.
He said: "Weeks and months ago, people hounded us to make expensive transfers.
"Then Coutinho comes and now the word is, 'How can they not start Muller?'
"But it was obvious that if [Coutinho] comes things would be difficult for Muller."
Mario Mandzukic of Juventus and Lyon’s Moussa Dembele have also been linked with a mid-season switch to the Red Devils as Ole Gunnar Solskjaer looks to bolster his attack.
United are struggling in the Premier League as they lie in 14th place after nine rounds with 10 points, leaving them just two points above the relegation zone.
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