The 25-year-old attacker has drawn interest from several Premier League clubs this offseason, including Manchester United as well as London rivals Chelsea and Arsenal.
Reports have also suggested that the Napoli attacker had previously declined an offer to move to the Saudi Pro League last year.
Gli Azzurri agreed to let Osimhen leave this summer, but the club is holding out for a team to meet his £110.8 million release clause.
So far, though, this price tag is proving too steep for European clubs, and could potentially force Napoli to lower their asking price to facilitate a sale.
TEAMtalk recently reported that the Italian giants may be willing to consider offers in the region of £85m to part ways with their in-demand attacker.
Conte addresses Osimhen future
In his first press conference as Napoli manager, Conte addressed the futures of several players, affirming that Khvicha Kvaratskhelia, Giovanni Di Lorenzo, and Stanislav Lobotka are not for sale.
However, a positive update regarding Osimhen's situation was not forthcoming.
Conte said: "As far as Osimhen is concerned, just as I knew very well about the situation of Kvara, Di Lorenzo, Anguissa, Lobotka, on which I have placed an absolute veto.
"Having said that, I know what the situation is regarding Osimhen. I know that there are agreements, that it is a situation different from all the others. I can only assist."
"He is a player of the highest level, exceptional, but I cannot enter into any discussion because it is part of previous agreements which have been clearly stated to me."
Arsenal rejoin race to secure Osimhen signature
Arsenal's sporting director Edu is believed to have prioritised signing a world-class number nine this summer after finishing two points behind Manchester City in the Premier League title race last term.
Mikel Arteta's current attacking duo, Gabriel Jesus and Eddie Nketiah, managed just nine goals between them last season, and the latter has been rumoured to be looking for a move away from the Emirates Stadium in search of regular playing time.
The Gunners were heavily linked with a move for Osimhen at the end of last season but began exploring other options due to the Nigerian's release clause.
Chelsea, on the other hand, are understood to be looking to bring in a new attacker to partner Nicolas Jackson, who had a mixed first season at Stamford Bridge.
However, bringing Osimhen to West London will likely require the Blues to sell a few of their existing players so as not to breach PL Profit and Sustainability rules.
The Blues were recently rumoured to have put academy products Trevor Chalobah, Conor Gallagher and Armando Broja up for sale this summer.