The 28-year-old has caught the eye of Sean Dyche as he looks to replace Michael Keane, following his big-money move to Everton.
The Ivory Coast international is believed to be available thus summer as he wants to play in the top flight and Sunderland are keen to boost new manager Simon Grayson's transfer kitty.
The Ivorian impressed after signing for the 'Black Cats' in January 2016 and was rewarded with a new £60,000 a week contract last year following interest from Everton, Crystal Palace and West Brom.
However his form dropped last season as Sunderland were relegated.
Kone made 31 appearances in all competitions for Sunderland last season.