Miguel Herrera has been fired as Mexico coach following allegations that he punched a television reporter.
Former Arsenal midfielder Abou Diaby has rejected an offer to join West Brom in favour of finding a new club in France.
The Italy forward has undergone a medical ahead of completing his move to the Grifoni.
"A new adventure begins and I have a great desire to get started," Matri said to Gazzetta dello Sport. "I'm looking forward to showing everyone that I am a different player to last year and that last year was only a blip on the journey."
The 29-year-old spent the second half of last season on loan at Fiorentina and scored four goals in 15 league appearances for the Viola having struggled to cement a place in the Milan side.
"I am honoured to be here," Matri said. "I want to start again and I am happy to be at the oldest club in Italy.
"I want to grow and be born again with this jersey.
"I chose Genoa for the shirt, for the history of this team and for its splendid fans.
"At the moment all I'm thinking about is doing well for Genoa, then maybe through my performances for Genoa I will get a call up from Italy."
Matri, who joined Milan last summer from Juventus, is under contract with the Rossoneri until June 2017.