The midfielder joined the Robins for an undisclosed fee from Oldham Athletic in the summer of 2013 and went on to make just under 200 appearances in all competitions during a six-year stay at Ashton Gate.
He has spent the last two seasons with Championship rivals Swansea City and received glowing praise from former City chief executive Mark Ashton following his departure.
Speaking via a statement released on the club website, he said: “Korey is our longest serving player and deserves huge respect for all he has done for this football club on and off the field.
“He has provided long-lasting memories and helped us improve and strengthen throughout his service - something we are all extremely grateful for.
“We wish him and his family all the best for the future.”
Smith has been without a club since leaving Swansea at the end of last season butit is now reported he is closing in on a deal to join recently relegated League One club Derby County.
Concerns over the Rams’ future have been swept away after they a takeover deal was secured by local property group Clowes Development.
The takeover sparked a flurry of new signings as the Pride Park outfit look to bounce back into the Championship under interim manager Liam Rosenior, who took charge following the departure of former manager Wayne Rooney.
The former Bristol City star has already added the likes of James Chester, Conor Hourihane and David McGoldrick to his squad and seems to be close to making Smith his eleventh signing of the summer.
The midfielder is said to be undergoing a medical with the Rams and could officially complete a deal to join the former Premier League outfit over the coming 24 hours.