Rafinha, 25, spent Tuesday in Munich undergoing medical examinations and has agreed a three-year contract with the Bundesliga club.
"This is a dream for me to be here," Rafinha told Munich newspaper tz after his visit to the club.
The Brazilian defender joined Genoa from Schalke last summer in a deal worth €8 million and made 34 appearances in his first season in Italy.
However, he has now been lured back to the Bundesliga, returning to the country where he impressed during his five years with Schalke.
The acquisition of Rafinha could end Bayern's interest in signing a new left-back as the Brazilian's arrival is likely to see Philipp Lahm move from right-back to the opposite side of defence.