Inzaghi has been at AC Milan since joining the club from Juventus in 2001.
Gattuso joined the Rossoneri in 1999.
Inzaghi told Milan’s official website: “The first and only thing that I want you to know forever is this: I played and won for us.
“Playing without sharing emotions means nothing, but you and I, we, we did everything together. We were always on the same wavelength, and no-one will ever take that away from us.”
Midfielder Gattuso added: “I’m going away for a year, this is goodbye. The moment has come to move myself away, but I don’t know yet where I will go.”