Coman was nearing the end of a two-year loan deal from the Italian champions, and Bayern have exercised an option in the deal to make the move permanent.
"Kingsley Coman is an important building block for the future of our team, so we decided to take the option," Bayern chief executive Karl-Heinz Rummenigge said on the club's website.
"Kingsley is a hopeful player with great potential. We are convinced that he will help us in the coming years."
Read the full article on eurosport.co.uk: Bayern sign Coman on permanent deal