The Frenchman has been linked with a Goodison Park switch for much of the summer and was even reported to be on Merseyside earlier this month.
Last week, The Daily Mail reported that the Toffees had tabled a £14million bid and that Kone had submitted a transfer request at the Stadium of Light.
But the same publication now claim David Moyes has told Kone he’ll be going nowhere this summer, because Sunderland can’t attract a replacement of similar or higher quality.
The majority of Moyes’ additions since being appointed Sunderland manager have been inexperienced youngsters with potential, such as Paddy McNair and Adnan Januzaj (on loan).
With Younes Kaboul moving to Watford and John O’Shea out injured, selling Kone would leave a gaping hole in Sunderland’s defence – and the Black Cats are struggling to attract signings that can improve upon their current squad.
Speaking at his pre-match press conference ahead of a League Cup tie with Shrewsbury, Moyes said; “I’ve had that offer (from Everton) for a while now and I could have invested that money two weeks ago, but that’s very difficult.
“It’s very difficult to invest in players given the prices – in terms of who you can get for your money. It will be very difficult to replace Lamine and until we were able to get close to a suitable replacement, I would never do the deal.
“So yes (the offer has been rejected) by me. I have said that Kone is staying.”