The Reds were continuously linked with the 23-year-old in the summer, with one report claiming that they made an enquiry to establish Napoli's asking price and the player's wage demands.
However, a summer switch to Anfield ultimately failed to materialise for Kvaratskhelia after new Napoli boss Antonio Conte insisted that the Georgia international would not be sold.
Liverpool eventually turned their attention to other targets towards the end of the summer transfer window and opted to swoop for Federico Chiesa, who has move to Merseyside from Juventus for an initial £10m.
It is thought that Liverpool are still admirers of Kvaratskhelia, though, and their alleged interest comes at a time when the future of star attack Mohamed Salah is believed to be uncertain having now entered the final year of his contract at Anfield.
The likes of Real Madrid, Barcelona, Paris Saint-Germain, Manchester City, Manchester United, Arsenal and Newcastle United have also been credited with an interest in Kvaratskhelia.
In-demand Kvaratskhelia to commit future to Napoli?
However, Napoli's sporting director Giovanni Manna has suggested that Kvaratskhelia's long-term future will be with his current club despite interest from other European teams.
Speaking to Sky Sport Italia, Manna said: "[Kvaratskhelia] still has three years on his contract. There have been teams approaching him, but we've shared the desire to continue together.
"We're working with the player's entourage to reach a contract renewal. We're calm, there's no need to rush."
Kvaratskhelia has developed into a star player for Napoli since joining the club from Dinamo Batumi in the summer of 2022, scoring 27 goals and providing 28 assists in 93 appearances across all competitions.
The four-time Georgian Player of the Year was named in the 2022-23 Serie A Team of the Year after contributing with 12 goals and 13 assists in 34 games to help Napoli win the Scudetto.
He followed up that impressive campaign by scoring 11 goals and registering eight assists in 34 Serie A matches last season, and has since made a bright start to the current campaign, recording two goals and two assists in four Italian top-flight games.