Tapalovic is accused of “leaking” private conversations between coaching staff to Neuer.
According to the report, Neuer, 36, would then relay information to other senior figures within the playing squad.
Following his departure, Germany goalkeeper Tapalovic posted an emotional farewell on Instagram.
“Dear Toni, today is the end of an era at FC Bayern Munich,” he wrote.
“With you after 11.5 years, not only an absolute pioneer of the modern goalkeeping game, but above all a great person is leaving the club.
“Everyone, not only in Munich, knows that all these achievements would never have been possible without you!
“Last but not least, you shaped and lifted me and my goalkeeping game to a new level. I will miss you!”
Neuer’s post received heart emoji comments from former Bayern stars David Alaba and Franck Ribery.
Tapalovic served as best man at Neuer’s wedding to ex-wife Nina Weiss in 2017.
After suffering an injury in the Champions League while playing for Bayern that year, the goalkeeper walked down the aisle on crutches.
After breaking his leg in a skiing accident last month, Neuer was ruled out for the rest of the season.
Bayern has since signed Swiss No. 1 Yann Sommer to deputize.