The Blues are keen on the Sunderland defender this summer.
The Blues are in the process of rebuilding their defence before the start of the season after selling John Stones to Manchester City.
The youngster completed a big money move to the Etihad Stadium on Tuesday, sparking the Toffees into life in the transfer window.
They have already secured the signing of Ashley Williams from Swansea City but are still in the market for another defender.
Kone appears to be top of their wish list, with the Daily Mail reporting they have already seen an £18 million offer rejected.
That has not deterred manager Ronald Koeman though, as the Liverpool Echo claims that he is confident of securing a deal for the defender by the end of the week.
Sunderland are not keen to lose their man after he starred for them in the second half of the campaign.
However, with the money being offered they will sanction a sale, although not until David Moyes has found a replacement for the 27-year-old.
Everton manager Ronaldo Koeman