The Germany international, who tasted World Cup glory in Brazil over the summer, is reported to have been identified as a top target by Louis van Gaal after he inherited the managerial reins at Old Trafford.
An approach was made, with United eager to bring world-class talent into their ranks as they seek a reversal in fortune.
Muller admits a move would have been of financial benefit to him, with the money on offer "astronomical".
He has, however, taken a decision he believes will be of greater benefit to his long-term career, with the 24-year-old happy and settled at Bayern.
Muller has spent his entire career to date with the reigning Bundesliga champions and sees no reason why he should turn his back on the club to go chasing a bigger pay packet elsewhere.
He told Bild when asked if United had come close to signing him: "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."