The Blues have been pursuing Osimhen's signature all summer long, although a deal initially proved unfeasible as Napoli demanded that his release clause of circa £113m was paid in full.
However, with Osimhen's desire to leave now out in the open, and the Nigeria international now frozen out of Antonio Conte's squad, Napoli have relented on their previous price demands.
With Romelu Lukaku on the verge of joining Napoli in a deal that could fetch up to €45m (£38.1m), the door was seemingly open for Osimhen to head to Stamford Bridge, even if a swap or part-exchange involving Lukaku did not come off.
The Blues are yet to strike a deal with Napoli, though, and according to journalist Fabrizio Romano, Saudi Pro League side Al-Ahli are now close to agreeing a fee with their Italian counterparts.
Al-Ahli, Osimhen 'yet to agree personal terms'
Romano claims that the 2022-23 Serie A champions are prepared to accept a €65m (£55.1m) package, and the accord between the clubs is "almost done", with only a few small details to be ironed out.
However, Al-Ahli are still a ways away from reaching an agreement with Osimhen on personal terms, and there are apparently two key conditions needed for the Nigeria international to commit his future to the Saudi titans.
As well as an "important" salary, the former Lille man will supposedly demand that a release clause also be inserted into his contract, allowing him a seamless return to Europe should he wish to play in the Champions League again in the future.
The 25-year-old has produced 76 goals and 18 assists in 133 appearances for Napoli over the past four years, but he has not made an appearance for Conte's side this season due to the ongoing transfer uncertainty.
Osimhen signed a new contract with Napoli in December 2023, although those terms only run until 2026 and were agreed primarily to protect his transfer value with a nine-figure release clause rather than being an indication of his long-term future.
Will Chelsea sign a striker if they miss out on Osimhen?
With Lukaku desperate to reunite with Conte, Conte desperate to reunite with Lukaku and Osimhen seemingly being the missing piece of the Chelsea puzzle, all of the signs were pointing towards the Belgian and Nigerian swapping dressing rooms.
However, the significant finances involved in bringing Osimhen to West London have to be factored into the equation, and a separate report from The Sun claims that he would want a staggering £500,000-a-week salary to join the Blues.
Al-Ahli and Paris Saint-Germain - also looking for a new striker after Goncalo Ramos's injury - would be better-placed to bow to Osimhen's demands, and with just a few days of the English window remaining, a deal appears to be virtually dead in the water.
One other realistic option for Chelsea would be Jonathan David, the man who replaced Osimhen at the tip of the Lille attack, as the Canadian is out of contract in 2025 and could be sold for between £20m and £30m this summer.
Having witnessed his side hit Wolverhampton Wanderers for six on Sunday, though, Enzo Maresca may feel that a new number nine is not necessary, even though the absence of a 20+ goal-a-season striker has often stuck out like a sore thumb in West London.