Hull boss Steve Bruce has identified Cole as one of his top targets as he looks to bolster his attacking options this summer.
Bruce is thought to have held talks with free agent Cole following his departure from West Ham earlier this summer.
A number of other Premier League clubs are believed to be keen on Cole, with the likes of West Brom and Crystal Palace credited with an interest in the former Chelsea man.
Cole is also attracting interest from abroad, with the 29-year-old understood to be talking to Turkish clubs, including Besiktas this week about a move.
However, Bruce is keen to win the race for Cole, with his need for forward reinforcements increasing after seeing a deal for Burnley's Charlie Austin fall through.
Cole is expected to make a decision on his future later this week once he hold talks with his Turkish suitors with the player keen to fix himself at a new club.