Conte simply doesn't want him to leave as he sees Kvaratskhelia as an important part of his system. This is why he is making statements like these.
However Kvaratskhelia made a statement like this after Conte's words.
https://x.com/FabrizioRomano/status/1806098263021416745?t=08Ovz3J5buTjMzeVPQ-u_A&s=19Things are getting heated though.

He doesn't want Conte to talk on his future clearly. Therefore I believe things are still likely to change after Euro 2024. Teams would continue to be interested in him until then.
It seems to me that even the first half of Georgia’s game against Spain is enough to understand that Kvartskhelia deserves better than Napoli (a club without the Champions League next season). I don’t know which clubs his future may be connected with, but purely visually he suits Liverpool very well, a fast, energetic player. Judging by Kvaratskhelia’s own comment, he already has options other than Napoli.