The 40-year-old sustained the injury during Saturday's 2-1 victory away from home, with the club confirming the diagnosis following an examination by their medical department.
Neuer was replaced shortly before half-time after experiencing discomfort in his right foot. Jonas Urbig came on in his place and is expected to take over in goal while the Bayern Munich captain recovers.
The club have not provided a definitive return date, describing Neuer as unavailable for the time being. However, reports in Germany suggest the injury could keep him out for around two months, potentially ruling him out until after the winter break.
The setback comes after Neuer had already missed several matches because of problems with the same foot. His latest appearance had also been a landmark occasion, as he became Bayern's oldest player in Bundesliga history at 40 years and 225 days.
The goalkeeper's afternoon in Mainz had begun with a record but ended prematurely. He was unable to continue after the injury and was replaced by Urbig, who has increasingly been involved in the first team as Bayern manage their veteran captain's workload.
Bayern's victory was secured by goals from Joshua Kimmich and Lennart Karl, with Harry Kane providing an assist. The result maintained their position at the top of the Bundesliga, but the loss of Neuer represents a significant concern heading into the international break.
Urbig now has the opportunity to build on his recent appearances and provide cover for one of the most experienced goalkeepers in the game. The 23-year-old has already featured regularly this season and is familiar with the demands of stepping in when Neuer is unavailable.
The precise length of the captain's absence will depend on his recovery, with Bayern yet to confirm when he can resume full training. For now, the priority will be allowing the injury to heal properly rather than setting a premature return date.
Neuer's experience and leadership remain important to the German champions, but they will have to manage the coming weeks without their first-choice goalkeeper.