The club had previously failed with an approach for Toure earlier in the summer, with the player unsure about the move.
But now after talks with Reds boss Brendan Rodgers, it appears that Toure is now warming to the idea and Trabzonspor president Ibrahim Haciosmanoglu claims a deal is close to beind done.
"We have agreed terms with Kolo Toure and we will bring him to the club," he told Hurriyet.
Liverpool are ready to let Toure - who joined last summer on a free transfer from Manchester City, penning a two-year deal - leave the club, having bolstered their defensive options in recent weeks with the signing of Dejan Lovren from Southampton.
Ambitious Trabzonspor are hoping 33-year-old Toure will be joined by Turkey international striker Mevlut Erding, and they are locked in talks with Saint-Etienne over his signature.