Press Association Sport understands the 27-year-old central defender has been on Merseyside – a photograph of him posing with a Toffees fan has emerged in the last 24 hours – with a view to pushing through an £18million switch.
Everton sold John Stones to Manchester City in a British record £47.5million transfer for a defender on Tuesday, and are understood to have agreed a fee with Swansea for Ashley Williams.
However, they have been in touch with the Black Cats over Kone and are aware what they would have to pay to land the Ivory Coast international, but there is no formal deal in place.
The Paris-born defender has been a major hit on Wearside since his £6million switch from French club Lorient in January, playing a key role in Sam Allardyce’s rescue mission as Sunderland managed to stave off relegation from the Premier League for a fourth successive season.
It is understood discussions took place over an improved contract with Kone happy at the club, but a new deal has subsequently failed to materialise and with Allardyce having now departed to take up the England job and been replaced by David Moyes, the situation remains unresolved.
Everton had taken advantage of that uncertainty by making their interest known, but they have as yet not pushed home their interest an it is understood several other clubs, alerted to the player’s possible availability, are now monitoring his situation closely.
Kone is due to return to training at the Academy of Light on Wednesday with the club’s fans desperate for him to stay despite the £8million capture of Chelsea’s Papy Djilobodji.