Luca Toni is seen as key in the deal, with the big centre-forward a close friend of Italy teammate Gattuso.
Bayern president Franz Beckenbauer declared: "Gattuso is perfect. If he would leave Milan, he would be an important signing for our team."
On interest from Milan in Toni, Beckenbauer added: "Toni has enabled us to win so much this season and we will not sell."