The Serie A side is now in the driving seat for the 21-year-old, who enjoyed a brilliant debut season at Lille.
Osimhen has been linked with several big clubs in Italy, Spain and England, but he could now be set to join the Naples outfit as the Naples now confident they have won the race to sign the Nigerian forward and his teammate at Lille of France, Gabriel for a combined price of 100 million.
Sources disclosed that Osimhen and Gabriel travelled from Lille to Italy where they have started talks with Aurelio De Laurentiis, the Napoli owner and his sporting director, Cristiano Giuntoli.
The offer made by the Serie A club is split 70m for Osimhen and 30m for Gabriel, which meets Lilles asking price.
The two players are set to stay in the south of Italy for a few days, where they will hold talks with Napoli manager Gennaro Gattuso and a deal could be finalised quickly.
Nigeria international Osimhen only joined Lille last summer from Charleroi for just 12m but had an excellent season in France with 13 league goals.
Sources have told ESPN that he would prefer a move to the Premier League where he dreams of playing. Chelsea and Manchester United are two clubs interested in him but are yet to make an offer. However, sources have told ESPN that Napoli are confident they can convince him to move to Italy.
Brazil under-23 international Gabriel has also attracted a lot of interest, especially from England. Everton and Chelsea have both spoken to his agents but he is keen to move to Italy.