The 26-year-old has been a key feature of the Napoli side that have romped to a 19 point lead at the top of Serie A and reached the Champions League quarter-finals.
It's reported that president Aurelio de Laurentiis, however, would command a big fee after the sale of Kalidou Koulibaly to Chelsea summer.
It's said that de Laurentiis feels he sold Koulibaly too cheaply and wants to avoid doing the same again with Kim by letting his contract run down, so would sell for a large fee this year.
According to Get Football News Italy, the defender is likely to move to one of Europe's top clubs, with Manchester United said to be interested.
It had been reported that Kim has a release clause of just under £44million, but it is now believed that the clause depends on who the South Korean would join, with it said to be varying from £48.3million to £61.5million.
Kouliably left Napoli for Chelsea for £33million with a year left on his deal, meaning the Italian side were almost forced to sell for a cheaper fee.
De Laurentiis is said to be unwilling to run that risk again with Kim, so could part ways with the defender this summer.
The player did, however, sign a five-year contract in the summer.
It's unlikely any Italian sides would be able to afford the required fee, meaning a move to Europe is likely.
United have reportedly already made contract with Kim's representatives as they look to strengthen their defensive ranks in the summer.
The player himself, however, told You Quiz on the Block in December that he was 'disturbed' by speculation around his future.
He said as cited by Sports Central: 'There are a lot of transfer stories around me, but I’ve only been at Napoli for six months. 'I’ve avoided interviews, and one of the reasons is that the rumours about me disturb me.
He also recently said rumours linking him with a move to United 'are not true'.
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