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Muller has spent his entire career with the reigning Bundesliga champions but has found his game time severely limited this season.
The 34-year-old has started just four league games so far this season and was left on the bench for the team’s 1-0 win against FC Koln on Friday.
The Germany international’s contract runs out at the end of the season but, according to Bild, it is unlikely that he will sign a new deal.
Muller has been vocal in his desire to continue playing professionally beyond this season and reports suggest that United are one of the teams ready to offer the attacker a contract in the summer.
Erik ten Hag is said to be a long-time admirer of the veteran German having reportedly failed with a move for Muller last January.
Bayern boss Thomas Tuchel has not publicly commented on the rumours but has reaffirmed his desire for Muller to see out his career at Bayern.
‘Of course we want him on our team,’ Tuchel said earlier in the month. ‘It goes without saying that he wants to continue.
‘He is in great shape, keen to train and play, he is a footballer through and through. Why shouldn’t he continue playing at the highest level?
‘It’s only November now, so of course we can’t say how his role will develop, but he is a legendary player, an icon for our team and gets the maximum respect.’
United are set to host Bayern in the middle of December in their final Champions League group-stage fixture.
Ten Hag’s side currently sit bottom of Group A and face a must-win encounter away at Galatasaray first.
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