Everton's Phil Jagielka
The Blues are on the hunt for a central defender before the start of the defender after John Stones completed a protracted move to Manchester City.
They have identified a Black Cats star Kone as one of the men they want, with the Daily Mail reporting that they have made an £18 million bid and the player is on Merseyside ahead of a medical.
Manchester United's Marcus Rashford in action with Everton's Phil Jagielka
However, no deal between the two clubs has been reached yet as David Moyes wants a replacement lined up before sanctioning the exit.
According to the Daily Mail, he was hoping to try and secure Everton players as part of the deal, with Phil Jagielka chief among his thoughts.
Everton's Phil Jagielka applauds the fans after the game
The Toffees have rejected that request though and will not entertain the idea of their captain leaving this summer, especially given they have already lost Stones.
There is every chance that Jagielka could seek a move given first team football is likely to be hard to be hard to come by this season.
Everton's Phil Jagielka looks dejected after Manchester United's second goal