The 30-year-old has agreed a deal with the Serie A side that is reported to run until the end of the season, with the option to extend it by a further year.
The forward will be officially presented as a Genoa player at a news conference on Tuesday.
However, the Italy international's career has been blighted by knee injuries and he suffered another cruciate ligament rupture in April.
Rossi could make his debut against Atalanta on December 11 and will be able to watch his side in action on Monday against Verona.
Genoa sit 17th in the table but have claimed four points from their last two league games, which included a 1-1 draw against Roma.