Muller, who has spent his entire senior career at Allianz-Arena, has been a target for United over the past couple of years and Premier League outfit failed to persuade the forward to join them back in 2015.
The Red Devils have been keeping a close eye on the Germany international ever since and it appears the club are becoming increasingly confident they can wrap up a deal in the summer.
Bayern manager Carlo Ancelotti brought in Real Madrid's James Rodriguez on a two-year loan deal in the recent transfer window and this has added to the competition for places in the Bavarians' first team.
Muller, who recently turned 28, is no longer rubber-stamped to start for the German champions and the reduced playing time is already said to be becoming a real frustration.
A change in scenery may be the best option for the forward and United, who have made a good start to the season, and ready to swoop in with a sizable offer at the end of the campaign.