The 27-year-old's future has come under scrutiny in recent months following his poorest goalscoring season since breaking into the first team at the Allianz Arena, netting just five times in 29 Bundesliga appearances.
Tuttosport claims that he is increasingly unhappy with his reduced role under Bayern boss Carlo Ancelotti and is beginning to consider a move away from the club he has spent his entire career with.
United have also held a long-standing interest in Muller, seeing a world-record bid for him turned down in 2015, and reports now suggest that they have reignited their pursuit.
However, United will likely face competition from their Premier League rivals and Italian champions Juventus for Muller's signature should Bayern decide to sell.
Muller has scored 161 goals in 398 appearances for Bayern, helping the club to 17 trophies along the way.