The 23-year-old has gone from strength to strength under Luciano Spalletti since arriving from Fenerbahce in the summer to replace Chelsea-bound Kalidou Koulibaly.
Kim has played the full 90 minutes in six of his side's seven Serie A games so far - scoring twice - and he has started both of their Champions League clashes with Liverpool and Rangers.
With Kim in the team, Napoli have risen to the top of their Champions League group with 4-1 and 3-0 wins over the Reds and the Gers respectively, as well as making an unbeaten start domestically.
The South Korea international has contributed to two clean sheets in Serie A, as Spalletti's side sit at the top of the table - ahead of second-placed Atalanta BC on goal difference.
The 6ft 3in defender signed a three-year deal upon his arrival at Napoli until the summer of 2025, although both parties reportedly possess the option to extend his stay until 2027.
However, Gianluca Di Marzio reports that Kim could leave Napoli if a €50m (£44.1m) release clause is triggered, but teams will not be able to activate it during the January window.
Instead, clubs must wait until the summer before battling for the South Korean's signature, and the report adds that several teams have been alerted to his performances this term.
Everton were linked with a move for Kim before he elected to sign for Napoli, and Man United are now said to be keeping tabs on his situation in Italy.
Erik ten Hag successfully convinced Lisandro Martinez to follow him to Old Trafford from Ajax, and the Argentine has overcome a tricky start to form an effective partnership with Raphael Varane.
Martinez and Varane's performances have forced Harry Maguire and Victor Lindelof to accept spots on the bench in recent weeks, while Eric Bailly has joined Marseille on loan and Phil Jones has been ostracized from the squad.
Maguire still holds the captain's armband at Old Trafford, but he has not started in any of their Premier League wins this season after shipping six goals to Brentford and Brighton & Hove Albion in the opening two gameweeks of the season.
However, the 29-year-old has retained the faith of England manager Gareth Southgate and is expected to start for the Three Lions when they face Italy in the Nations League on Friday night.
Man United have risen to fifth in the Premier League table courtesy of their four-game winning streak and are back in action against locals rivals Manchester City in the Manchester derby on October 2 at the Etihad Stadium.