In my opinion, Osimhen will relatively stay with Napoli because so far no team is really serious about signing him other than just saying they are interested. Napoli will have to keep their star players with them for next season if they want to strengthen themselves as the strongest team in Serie A next season.
PSG is indeed eyeing the Napoli striker as an alternative player to replace Mbappe. However, they are aware that this player actually has no desire to move, as well as Napoli who do not want to let go of the player who brought him in and contributed greatly to winning the Italian Serie A last season. Because of that, Napoli provides quite a fantastic price tag for clubs that want Oshimen. So this might make other clubs not focus anymore on these forward players.
However, if in this transfer market PSG really loses Mbappe, it is likely that PSG will move quickly to try to redeem Osimhen from Napoli. In fact, the Napoli boss himself said about Osimhen and PSG. Namely Aurelio De Laurentiis said that only PSG would be able to pay the ransom for the best Napoli striker (Osimhen). And right now, as we know, PSG is trying to get Mbappe closer to clubs who are interested and want to make up for the transfer price offered by PSG. Even one of the Arab clubs seems to have made a high offer for Mbappe. and it seems that PSG will not prevent Mbappe from going there either. But everyone thinks that Mbappe will not be interested in Arab clubs. And in fact Mbappe still wants to finish his contract at PSG. I suspect that Mbappe has actually made an internal agreement with Real Madrid to join next season after his contract expires. because right now Real Madrid looks so relaxed and in no hurry to get Mbappe.
However, even though one time PSG made a high offer to Osimhen, the decision was still in Osimhen's hands. whether he will stay at Napoli or go to PSG. I personally feel that Osimhen is at home in Napoli. In fact, he has received a promotion with a salary increase and was offered a contract extension. Which means Osimhen has really been spoiled at Napoli so there's no reason for him to leave a Club that really values him at the moment. And some time ago the Napoli boss also said that they had no intention of selling their best players in this transfer market.