The winger has undergone surgery and will be absent for several weeks, meaning he will miss the last 16 clash with the Gunners.
The France winger, 30, was absent from the Bavarians' 2-0 win over Nurnberg on Saturday with the injury after going under the knife Thursday in a bid to completely eradicate the problem.
"An examination on Monday revealed that the bruise, which had pressed on a nerve, was completely removed at surgery last Thursday," a statement on Bayern's website read. "After healing, the 30-year-old will begin exercise therapy later this week and a week later he will enter back into running training.
"This will mean he will miss the club's next few games, including the first leg of the Champions League clash with Arsenal."
The Champions League holders travel to the Emirates on February 19 before the return leg on March 12.