Paul Pogba became the world's most expensive player when United paid Juventus £89.3m for his services along with signings Eric Bailly (£30m) and Henrikh Mkhitaryan (£26.3m).
But according to Bayern Munich financial partner Jan-Christian Dreeson, United also made a huge bid for Germany international Muller.
Manchester United reportedly made an £85million bid for Thomas Muller
Jose Mourinho is expected to be given another huge transfer kitty this summer
“There really was such a thing [an offer of more than €100m for Muller from United],” Dreesen told SportBild.
“Then came a fax from England. For us, however, a sale of Thomas Muller was never discussed.
“If a player fits us, we would be stupid to give him a short-term prospect of a record profit and to weaken him.”
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Who could Jose Mourinho target in the summer?
Mourinho is likely to add more firepower to his squad in the summer transfer window to lighten the load on Zlatan Ibrahimovic.
Antoine Griezmann has been strongly linked with a move to the Premier League but United may be keen to return with another bid for Muller this summer.
The 27-year-old has been at Munich since breaking through the youth system in 2008.