The Turkish Super Lig giants signed the defender last year for €3m and he has done enough in his first season in Europe to attract interest from the big leagues.
He has made 39 appearances for Fenerbahce but could be on his way out of the Turkish club in the upcoming transfer window.
Everton have been keeping tabs on him and it has been claimed Toffees scouts watched him twice in action for Fenerbahce this season.
The Merseyside outfit like him and according to Turkish outlet Ajansspor, the club are prepared to table an offer for him.
Frank Lampard wants Everton to move swiftly in the market this summer after narrowly avoiding relegation from the Premier League.
Kim is a player he wants and Everton are ready to put in a bid to snare him away from Fenerbahce.
He has a €23m release clause in his contract and it remains to be seen whether Everton are looking to trigger that.
The South Korean is keen on the move and wants to test himself in the Premier League from next season onwards.
