The Stadio Artemio Franchi side are ramping up their interest in the Uruguayan, with newly-appointed Viola boss Vincenzo Montella looking to add some experience to his side
The 28-year-old has been a mainstay in the Partenopei team since joining the club in 2007, and La Viola feel the Uruguay international's combative style will add some bite to their midfield.
The former Danubio man has only recently renewed his Azzurri contract until 2015, and only a substantial bid is likely to sway coach Walter Mazzarri into selling one of his prized assets.
The Tuscan club, who recently appointed Vincenzo Montella as their new coach, have had a busy summer so far, signing Matias Fernandez, Cristiano Lupatelli and Juan Cuadrado, as well as off-loading Valon Behrami, Alessandro Gamberini and Cesare Natali.
Fiorentina finished the 2011-12 Serie A season in 13th place, while Gargano helped Napoli to a fifth-placed finish and victory over Juventus in the Coppa Italia.