The Mackems defender is wanted by the Toffees as John Stones' replacement.
Kone scoring against Everton in May
The Toffees are expected to lose John Stones to Manchester City and have targeted the Ivorian as a replacement.
Everton's Stones is thought to be Manchester City bound
The report claims that Kone, who arrived at the Stadium of Light from Lorient in January, was interested in the switch, however, Everton will now go for Kara Mbodj instead.
The Anderlecht defender would have reportedly joined West Ham a year ago but for work permit problems, which it is claimed have since been resolved.
Everton have added just Maarten Stekelenburg to their senior ranks so far, though Aston Villa midfielder Idrissa Gueye is expected to follow in the near future.
Gueye (right) is set for a move to Goodison Park
Sunderland paid a reported £6 million for Kone, who steadied the Mackems' back line as they survived relegation at Newcastle United's expense.
The 27-year-old also weighed in with two goals in 15 top-flight appearances - two of which came against Everton in the 3-0 win which secured the club's safety in May.
Sunderland, meanwhile, are yet to make any new signings, with new boss David Moyes said to still be assessing the squad he inherited from Sam Allardyce last week (Mail).
New Sunderland boss Moyes