Roberto Martinez spoke of his desire to sign 2-3 more players - including a defender - before the transfer window shuts on September 1.
Everton currently have just two senior centre backs in Phil Jagielka and John Stones, with the latter the subject of intense interest from Chelsea, so Martinez desperately needs to add to his defensive ranks.
Koulibaly only joined Napoli last summer and has four years left on his deal, so it would take a significant transfer fee to persuade the club to do business.
Premier League rivals Southampton have also been linked with a move for the 24-year-old, but it appears Everton have made the first move.
Koulibaly was a regular for Napoli last season, making 27 appearances after joining from Belgian side Genk. The powerful defender has also been capped at Under-20 level by France.
He routinely bossed games single-handedly on the back line, and took a number of very good strikers completely out of matches. His potential is still quite high compared to what we've seen, and losing him would be a huge blow to a defensive unit that cannot afford such a loss.
Please don't make this deal, Napoli. Don't take our Kalidou away from us. You need him, the fans need him, the world needs him. Make the right decision.