The Frenchman was a key influence at the back as the Black Cats staved off relegation last season, making 15 top flight appearances after arriving from Lorient in January.
Everton have been strongly linked with the 6 foot 1 defender this summer, with Kone reportedly even travelling to Merseyside in a bid to push through a transfer deal earlier this month.
But David Moyes was quick to insist Sunderland would offer the 27-year-old an improved contract, maintaining a promise made by his predecessor Sam Allardyce.
And according to The Sun, Kone has now been offered a new deal, although his current terms aren’t due to expire until summer 2020.
The British red top further add that the Stadium of Light outfit have rejected a £14million bid from Ronald Koeman’s side.
Sunderland face Middlesbrough in a Premier League derby this weekend, whilst Everton face West Brom at the Hawthorns.