In the offseason I only saw rumors that PSG were interested in him, but Conte refused to let him go because he needed a good attack and knew that Osimhen was leaving Napoli. I doubt that Real wanted to buy Kvaratskhelia, because they already had Rodrygo and Vinicius, and they also knew that Mbappe was joining the team, so they would not have had a place for Kvaratskhelia, or one of the forwards would have had to sit on the bench, and I suppose it could have been Kvaratskhelia.
We know the closeness of elite teams towards ensuring they secure the bids of these big names. Antonio Conte is one of the best coach to handle Napoli this season, knowing the velocity to exert for rapid growth of the club. He's taking his managerial role really serious because there are major sets of goals to be accomplished. Napoli would release the Georgian to the potential bid this season? I don't think so because there's more exploring opportunities that await Khvicha Kvaratskhelia in the club. He's currently the best player in the club, he wouldn't settle for lesser outcome, other than thrilling for excellence in games.