The 22-year-old was already regarded as a player with a bright future during his time with Lokomotiv Moscow and Rubin Kazan, although he left Russia for Georgian side Dinamo Batumi following the invasion of Ukraine.
Kvartskhelia spent just a few months back in his homeland before Napoli snapped him up for just €13.3m (£11.5m) in the summer of 2022, and he has taken Serie A by storm since joining the Partenopei.
Forming a deadly attacking partnership with Victor Osimhen, Kvaratskhelia ended his debut campaign in Naples with a Serie A winners' medal around his neck and also helped his side reach the quarter-finals of the Champions League.
With 12 goals and 13 assists in the 2022-23 Serie A season, the winger set up more goals than anyone else in the division, won three player of the month awards and was named the league's Most Valuable Player.
The individual accolades continued to stack up for the Georgia international, who was named the 2022-23 Champions League Young Player of the Season and was nominated for the 2023 Ballon d'Or.
Kvaratskhelia has not had the same phenomenal impact in the early stages of the 2023-24 season amid Napoli's collective struggles, but he still boasts a respectable four goals and five assists from 17 matches in all tournaments.
The 22-year-old's exploits have naturally sparked rumours of a possible move away from Serie A, with all of Manchester City, Arsenal, Newcastle United and Manchester United said to be keen to bring him to the Premier League.
However, the English quartet's hopes of a deal have been harmed by an admission from Kvaratskhelia's father Badri Kvaratskhelia, who believes that his son is yearning to join Real Madrid at some stage in his career.
Speaking to Georgian football outlet Geo Team after the Napoli man played the full 90 of his side's 4-2 loss to Real in the Champions League on Wednesday, Badri Kvaratskhelia said: "For Khvicha, playing against Real Madrid was special.
"He has always dreamed and I am convinced that he still dreams of playing for Real Madrid. In my family, everyone is a fan of Los Blancos - except for me. For this reason it was a very special match."
Napoli are under little pressure to sell Kvaratskhelia, whose contract does not expire until the summer of 2027, but a bid of around €100m (£86.3m) could supposedly be taken into consideration.
Real boss Carlo Ancelotti is currently blessed with Brazilian duo Vinicius Junior and Rodrygo as his first-choice wingers, although the pair have been thrust into central roles this term to accommodate the Italian's system switch.
Vinicius is also facing two months out with a hamstring injury, and Real would have the funds necessary to win the race for Kvaratskhelia after allegedly ending their interest in Paris Saint-Germain's Kylian Mbappe.
Los Blancos have been frustrated in their efforts to sign the Frenchman for the past couple of years and are now thought to be prioritising younger targets who would not command such an astronomical wage or signing-on fee.