The Partenopei have been zeroing in on a new centre-back this summer after losing their star defender Kim Min-jae to Bayern Munich in a deal worth €50 million.
The South Korean international set the bar high after helping the Neapolitans win the Scudetto last season, and the Italian giants are assessing their options carefully.
Kilman emerged as a top target, but Napoli have already seen their offer of €35m turned down by Wolves.
Wolves’ plan to extend Kilman’s contract has seemingly slowed down Napoli’s pursuit.
However, the Serie A Champions are reluctant to give up the chase and are keen on recruiting him ahead of next season.
Despite not improving on their previous offer, Napoli are confident they will eventually agree a deal with Wolves.
The 26-year-old may also be open to a move as the Italians can offer him Champions League football.
Napoli are also working on a potential deal for RC Lens centre-back Kevin Danso.
Personal terms have been agreed with the Austrian defender, but they have had an offer worth €22m knocked back by the Ligue 1 club, who value him at €30m.
Napoli are also interested in Fiorentina midfielder Gaetano Castrovili. They are considering a swap deal involving Diego Demme.
Piotr Zielinski has been the subject of exit speculation this summer, but recent reports suggest he has decided to extend his contract with Napoli.
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