The report states that Alonso’s position at Bayern is under threat and that he may be open to a move away from the Bundesliga champions.
Galatasaray are on the lookout for a new defensive midfielder following reports linking Felipe Melo with a move to Inter Milan.
Alonso has previously played for Real Sociedad, Liverpool and Real Madrid. The experienced playmaker provided five assists and scored four goals in 42 appearances for Bayern last season.
The 33-year-old’s contract with Bayern expires at the end of the season.
Bayern won the Bundesliga by a ten goal margin last season and are once again favourites to lift the title this season.
Galatasaray won the league title for the 20th time last season and secured a place in the Champions League group stages.