The Nigeria international's desire to leave the Italian giants this summer is no secret, and he is believed to be at the heart of an intense two-way fight between the Blues and Les Parisiens.
Osimhen's Napoli contract - which he extended last year - includes a release clause worth around £113m, but the club are no longer insisting that the option needs to be paid in full for him to depart.
Instead, it was reported last week that Chelsea were close to agreeing a three-player swap deal, which would have seen Osimhen head to Chelsea and Romelu Lukaku and Cesare Casadei join Napoli, who would have also received £38m.
Napoli head coach Antonio Conte has made it his priority to reunite with Lukaku after their success at Inter Milan, although PSG are still in the race following a serious injury to Goncalo Ramos, who is out for three months with an ankle sprain.
Napoli, Osimhen 'working' on exit amid Chelsea interest
The French champions had cooled their interest before Ramos sustained his issue on the Ligue 1 opening weekend, and Osimhen is supposedly prepared to wait until the eleventh hour to join Luis Enrique's side.
Napoli boss Conte was loath to discuss the transfer talk when questioned by reporters last week, saying: "You all know full well the Osimhen situation was already there when I arrived and will be dealt with exclusively by the club.
"I did not get into the situation. You all know that the transfer market is very complicated and [the deal is] currently blocked. I cannot say any more than that."
Sporting director Manna was also asked for an update on developments by Sky Italia, to which he echoed Conte's sentiment by saying: "It's an objective fact, what the coach said, the situation is blocked," as quoted by the Evening Standard.
"Victor is an important player, with an important value and conditions. His desire is clear. We tried to maintain the integrity of the squad at this moment. We are not the only club in this situation and we'll see what happens over the next 10 days.
"At this time, it is not right to say what sensations there are. We are working with his entourage to find the best solution. As we said, he is an important player, so it's not that we can just find him anywhere [to play]. He is under contract for another two seasons, so clearly if we don't find a solution by August 30, we'll have to talk to the player."
Why are Chelsea pushing to sign Osimhen?
Ever since Todd Boehly's consortium took control of the club, Chelsea have been forking out eight-figure fees for a surfeit of unheralded names, but their lack of a prolific striker sticks out like a sore thumb.
Enzo Maresca's side had their moments in Sunday's Premier League opener against Manchester City, but Nicolas Jackson failed to make an impact, firing one brilliant chance at Ederson from point-blank range as the Blues lost 2-0.
Since touching down in Naples, Osimhen has had a direct hand in 94 goals from 133 Partenopei appearances - 76 of his own and 18 assists - winning the 2022-23 Serie A Capocannoniere prize with 26 efforts in 32 games.
The Nigerian's numbers reduced last season owing to Napoli's collective downfall, but he still managed 17 strikes in 32 games for an ailing side and would solve the Chelsea issue that should have been rectified some time ago.