The 29-year-old has been consistently linked with a move to the Premier League for a number of years now, with Manchester United and Manchester City thought to be most interested as they search for defensive reinforcements.
Paris Saint-Germain are also understood to be monitoring his situation, but no club has yet been willing to meet Napoli's €75m (£69.1m) asking price.
La Gazzetta dello Sport claims that Man City are currently leading the race but are not prepared to pay more than €70m (£64.5m), with talks further complicated by the frosty relationship between the two clubs following the collapse of a deal for Jorginho in 2018.
Napoli president Aurelio De Laurentiis has confirmed that Koulibaly will be allowed to leave this summer if the club receive a suitable offer, but both the club and the player would also still be happy if he stayed.
The Serie A outfit have set a deadline of this weekend to get any deal sorted due their league campaign getting underway against Parma on Sunday.