The German international player emerged as a target for Louis van Gaal after seeking reassurances of his role at the Allianz Arena.
Manchester United were confident they could prize him away for £25million but after discussions with boss Pep Guardiola, Muller penned a new five-year deal with the Bundesliga champions Bayern Munich.
And he has now claimed he spurned a lucrative offer from Old Trafford to do so, which would have seen him better his salary considerably.
When asked if United nearly signed him, the 24-year-old told German news provider Bild: "Of course. I was aware that I could earn more by making that transfer.
"The amount that foreign clubs were offering me was astronomical. But Bayern are my club, a change was not an issue for me."
Muller's reluctance to move was the start of a frustrating transfer window for the Red Devils.
Manchester United has seen Toni Kroos and Thomas Vermaelen snub interest to join Real Madrid and Barcelona respectively, while bids for Mats Hummels and Bastian Schweinsteiger have also been turned down.