The 24-year-old was in impressive form for his Italian side during the 2022-23 campaign, finding the back of the net on 31 occasions in 39 appearances, in addition to contributing five assists.
Osimhen netted 26 goals in Serie A, as Napoli won the title, while he managed five goals in six Champions League appearances, with the Blues making it to the quarter-finals.
The Nigeria international's future has been the subject of much speculation, with many of the leading clubs in world football said to be keen on his signature this summer.
Chelsea, Manchester United and Bayern Munich are said to be among his admirers, but Napoli president Aurelio de Laurentiis recently claimed that Paris Saint-Germain were the only club capable of affording him.
De Laurentiis said that only offers in the region of €200m (£170m) would be considered, with Osimhen set to remain with his Italian club this summer.
The Nigerian has another two years left to run on his contract, but Napoli are allegedly keen to sign him to another deal, as he will only have 12 months left next summer.
According to Sky Italia, there are ongoing negotiations between Osimhen's agent Roberto Calenda and De Laurentiis over a new deal in Naples.
However, the report claims that Napoli want a release clause of €200m (£170m), but Osimhen is asking for the buy-out fee to be set at €120m (£104m).
Osimhen made the move from Lille to Napoli in the summer of 2020, and he has represented the Italian outfit on 101 occasions, scoring 59 goals and registering 14 assists in the process.
Chelsea are no longer believed to be considering making a move for the forward this summer, while Man United have switched their attention to Atalanta BC's Rasmus Hojlund.
Bayern, who were also linked with Osimhen earlier this summer, are currently focusing their efforts on bringing Tottenham Hotspur's Harry Kane to Allianz Arena.
Napoli are currently building towards their Serie A opener against Frosinone on August 19, and they played their first pre-season friendly on Thursday, recording a 6-1 win over Anaune.
The Italian champions have made two signings during this summer's transfer window, bringing in Giovanni Simeone and Giacomo Raspadori on permanent deals after loan spells at the club.