Everton appointed Kone's former manager Sam Allardyce on Wednesday, and lead the early betting at 3/1 after Sky Bet opened a market on the Ivorian's next club.
The Toffees, along with West Brom, have long been linked with a move for Kone, whom Allardyce brought to England while in charge of Sunderland 22 months ago, although the trail seems to have gone cold in recent months.
Kone's form - his Sunderland side are currently battling relegation to the third tier - may be a factor in that, although it does not appear to have deterred West Ham, who were recently credited with interest in the 28-year-old after hiring another of his old bosses, David Moyes.
According to Sky Bet, West Ham, at 5/1, and West Brom, at 6/1, are also in with a shout of signing Kone this winter.
Swansea City (8/1) and Crystal Palace (10/1) are also listed as contenders, while the French-born player is 6/4 to remain at the Stadium of Light.
Could Kone leave Sunderland in January? And will Everton, West Ham, West Brom or someone else win the race for his signature?