The 21-year-old has been in impressive form for Napoli since arriving from Dinamo Batumi over the summer, scoring seven goals and registering eight assists in 15 appearances.
Kvaratskhelia has managed five goals and five assists in 11 Serie A appearances, while he has also netted twice and registered three assists in four Champions League outings.
The Georgia international penned a five-year deal with the Blues upon his arrival, but there is currently widespread speculation surrounding his future.
Manchester City are believed to be keen to sign the attacker next summer, while there have also been suggestions of interest from Real Madrid and Paris Saint-Germain.
Napoli are believed to be considering opening talks with the summer arrival about a new and improved contract due to the his impressive form.
According to Badri Kvaratskhelia, there have not yet been any talks over a new deal, but the in-demand forward currently has no desire to leave Naples.
"Nobody has spoken to us about the contract yet. At this stage, Khvicha doesn't even think about it, he only thinks about playing. Khvicha is informed about market rumours, but it doesn't influence him at all," he told Imedi TV.
Badri Kvaratskhelia also discussed his son's happiness at Napoli, who through to the last-16 stage of the Champions League and sit top of Serie A.
"When you are in the stands of the Napoli stadium and your surname is shouted by 60,000 fans, this feeling cannot be conveyed. Especially when it's a footballer coming from the Georgian league. I was always convinced that Khvicha would do well from the start," he added.
"He does not need an introduction or too many words, he has shown what he can do. There are no groups in the team, there is unity. You can feel a friendly attitude.
"Luciano Spalletti has a very good relationship with his players. Khvicha goes to training at 8am. Around 2pm he comes back and rests. He had a day off and we went for a walk in the city."
Kvaratskhelia played for Rubin Kazan between 2019 and 2022, but he departed the club earlier this year following Russia's invasion of Ukraine.
The Georgian scored twice and provided two assists in his recent two European appearances against Ajax, while he helped his team record a 1-0 victory over Roma in Serie A at the weekend.