From July 1, Kim has a €60 million (£51.6m) release clause, according to the source. This is what Manchester United have agreed to trigger, with Tuttomercatoweb claiming that the 20-time English champions have also agreed a €3m settlement to sort this deal quickly.
Kim only moved to Napoli last summer from Fenerbahce in a deal believed to be worth around €18m
The move for Kim will likely relegate Raphael Varane to being cover, with the Frenchman having suffered with injury issues during his time at Old Trafford. Harry Maguire is expected to leave the club.
Kim is valued at €50m by Transfermarkt.
More Manchester United stories
Manchester United transfer news is doing the rounds, with the Red Devils said to be in the market for several stars this summer.
United have also been reported to have interest in US midfielder Taylor Booth, while Adrian Rabiot has been linked, too.