United broke the world transfer record with their re-signing of Paul Pogba from Juventus for an initial fee of £89.3million ($145M) and big spending has been a particular feature under Jose Mourinho and his predecessor Louis van Gaal at Old Trafford.
Jan-Christian Dreesen, the executive board member who is oversees Bayern's financial affairs, told Sport Bild that Muller has also been on United's radar for a similar figure, even though the prospect of selling the 2014 World Cup winner was never seriously considered.
"There really was such a thing. A fax came from England," Dreesen said.
"For us, however, a sale of Thomas Muller was never discussed.
"If a player fits us, we would be stupid to take the short-term prospect of a record profit to weaken the team."
Muller is chasing a fifth consecutive Bundesliga title for Bayern this season but the 27-year-old has struggled for his best form under Carlo Ancelotti, scoring four goals in 26 appearances across all competitions and just one in Germany's top flight.