Hajrovic, who represented his country at the FIFA World Cup in Brazil, moves to Germany on a four-year deal and will be presented on Thursday.
The former Grasshoppers man, 22, only joined Galatasaray in January, making just 12 appearances for the club.
Bremen head coach Robin Dutt told the club's official website: "He is the additional option in attack, which we desired.
"He is quick, good on the ball and very strong when dribbling one on one."
Hajrovic may be looked upon to fill the void left by the departure of Aaron Hunt, who joined Wolfsburg earlier in the close season.