Genoa admit they considered Luis Enrique for their coaching position in the summer.
Enrique is battling to turn things around at AS Roma.
"Luis Enrique was suggested to us, but we couldn't reach an agreement with him," sporting director Stefano Capozucca told Forza Roma.
"I consider him to be a valid Coach and Roma made an excellent choice.
"Despite the initial difficulties, which were to be expected, Roma will return to the higher echelons of the League."