The Georgian winger has scored three league goals and provided an assist, while also finding the net twice in the Coppa Italia.
His performances have attracted attention from several clubs, with La Gazzetta dello Sport reporting that Napoli scouts have been monitoring him.
Kvernadze has already been compared with fellow Georgian Khvicha Kvaratskhelia, with the pair sharing the same agent, Mamuka Jugeli.
Napoli are not alone in watching the 22-year-old, with Roma and Fiorentina also reported to be interested.
Premier League interest could further complicate Frosinone's attempts to keep Kvernadze beyond January.
Hull City are also understood to be monitoring the winger, who joined Frosinone from Kolkheti Poti in 2023.
The initial move was a loan before Frosinone completed his permanent signing for less than €1m overall.
His value has subsequently risen through his performances in Serie A and Serie B, making Kvernadze an increasingly valuable asset for the Italian club.
Frosinone could therefore face a difficult decision in January if interest from Napoli, Roma, Fiorentina and England develops into concrete offers.