Napoli coach Antonio Conte confirmed the previous day that Kvaratskhelia - nicknamed "Kvaradona" by the fans in homage to Diego Maradona - had asked to leave the club, with Paris Saint-Germain the likely destination.
His teammates did not let the controversy affect them as they beat lowly Hellas Verona 2-0 on Sunday to move four points clear of Inter Milan after the defending champions moved into second place earlier with a 1-0 win at Venezia.
Atalanta were third after they were held 0-0 at Udinese on Saturday. Inter have played two fewer matches than Napoli and one less than Atalanta.
Kvaratskhelia was out of Sunday's match in Naples through injury anyway, saving Conte from perhaps a difficult choice.
Verona had beaten Napoli 3-0 in Conte's first league match in charge.
But there was no such danger this time as Napoli took the lead in the fifth minute. Giovanni Di Lorenzo played a one-two with Romelu Lukaku and then curled in a shot that hit the left post and went in off Lorenzo Montipò's back, going down as an own-goal from the Verona goalkeeper.
Frank Anguissa scored a spectacular goal in the 61st minute as he cut inside from the right, played a one-two with Lukaku and then drove the ball powerfully into the near side of the net from outside the area.
Inter Milan bounced back from a demoralising Italian Super Cup loss but it was harder than might have been expected at relegation-threatened Venezia.
Inter lost the Italian Super Cup final to city rivals AC Milan in Saudi Arabia on Monday after throwing away a 2-0 lead.
Inter took the lead in the 16th minute when Lautaro Martínez showed fantastic control on a long ball over the top. His volley was parried by Filip Stankovic but Matteo Darmian turned in the rebound.
Venezia almost levelled in the 74th but Gianluca Busio curled a fine attempt off the base of the far post and Inter held on for their sixth straight away victory without conceding a goal.
Venezia remained 19th in the 20-team standings, five points from safety.
New Genoa owner Dan Sucu was in the stands to see his team beat Parma 1-0 in a relegation fight, their first league home win of the season.
They needed a huge slice of luck as Morten Frendrup's strike took a huge deflection off Parma defender Enrico Delprato to give Genoa the lead in the 65th minute.
Genoa moved four points above Parma and five away from the relegation zone.
Roma forward Artem Dovbyk converted a penalty with the last kick of the match, in the eighth minute of stoppage time, to snatch a 2-2 draw at Bologna.
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