The Colombia international spent last season on loan at the San Paolo, making 24 appearances in all competitions.
Under the terms of the deal, Napoli would have been compelled to sign the player if he played 25 games for them last season.
They still have the option to sign him permanently for £3.1m but that agreement is due to expire this week.
Ospina himself is unsure if Napoli boss Carlo Ancelotti wants to keep him at the club due to him being dropped just short of the 25 games.
According to The Sun, the Gunners have told the Serie A side that they must decide whether they want to exercise the £3.1m purchase clause by Saturday.
The report adds that Arsenal believe Napoli want to sign the 30-year-old, but that they only want to pay £2m for him.
The Premier League club are unwilling to lower their valuation however and believe that other clubs will pay closer to £3m for him.
Unai Emery may even bring Ospina back to the Emirates after the retirement of Petr Cech at the end of last season.
Ospina could be utilised as back-up for Bernd Leno and even push him for the starting slot.
The stopper has made 70 appearances for Arsenal in his four seasons at the club between 2014 and 2018.
Elsewhere, it is being claimed that Emery is close to agreeing a deal to sign Saint Etienne defender William Saliba.
The Gunners have been monitoring the 18-year-old’s situation following a strong campaign for the Ligue 1 outfit.
And Le10 Sport are reporting that Arsenal are willing to pay the £26m asking fee – but may be forced into loaning the youngster back to France for another season.
The French publication claims both parties are still locked in negotiations.
However, a deal is still close enough that it could be finalised in the next few weeks.