The two clubs maintain an excellent rapport, exchanging numerous players in recent years.
According to Sky Sport Italia the Spaniard has refused to join the Grifone on a season-long loan.
Suso started Milan’s second Serie A game of the season against Empoli but failed to impress.
A host of Spanish sides have been linked with the 21 year-old, but all have been rejected.
Milan were hoping to use the opportunity to sign Diego Perotti, but not without Suso.
Genoa are in trouble, with a deal for Roma winger Juan Manuel Iturbe collapsing.
The Grifone may yet get their man, but Suso is determined to stay at Milan.
