The Germany international has been linked with a move away from the Allianz Arena this summer following an apparent training ground bust-up with Pep Guardiola.
Muller has failed to nail down a regular place in the Spaniard's starting XI and could be tempted by a move to the Premier League.
Reports earlier in the week suggested that United were plotting an audacious swap deal including misfit winger Angel Di Maria, but sports outlet Get French Football claims that Louis van Gaal has submitted a formal offer of £57m instead.
However, it is said that Bayern have rejected the bid as they have no interest in selling Muller.
GETTY
Thomas Muller has been linked with a move to Man Utd this summer
United are believed to be on the look-out for a new attacker following Radamel Falcao's return to Monaco and subsequent loan switch to Chelsea.
Robin van Persie is closing in on a permanent move to Fenerbahce, while Javier Hernandez is also expected to leave Old Trafford.
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