The four-year deal for the 21-year-old also includes a 50% sell-on clause.
As far as Bayern are concerned, their transfer deals for the 2022 summer window are done.
The priority for Bayern Munich bosses now is to find a suitable loan club for 18-year-old Gabriel Vidović.
The attacking midfielder has been linked with a move to fellow Bundesliga clubs Mainz, Bochum, and his birthplace Augsburg.
Bayern hope to get the Croatian U21 international used to the pace and style of Germany’s top division.
If a move to a fellow top-flight club looks unlikely, 2. Bundesliga side Jahn Regensburg and other European clubs, including teams from the Netherlands, Belgium and the English second division, are also said to be interested in the young prospect.