Mueller was shown the first straight red card of his career for kicking Ajax defender Nicolas Tagliafico in the head.
If Bayern's appeal is unsuccessful, the 29-year-old will miss Bayern's last 16 first leg against Liverpool at Anfield on 19 February and the second leg at the Allianz Arena on 13 March. Bayern hope to have Germany international for the home tie.
Mueller apologised following dismissal. "Of course I didn't mean to do it. It wasn't intentional," Mueller told reporters.
Mueller has five goals and six assists in 25 matches in all competitions this season.
The Ajax midfielder Maximilian Woeber was also sent off during the 3-3 draw in Amsterdam. He was expelled for a tackle on Bayern's Leon Goretzka.
Woeber will be absent for both of Ajax's last 16 clashes against the holders Real Madrid.