The Ivory Coast international completed his switch after signing a deal at Liverpool's Melwood training ground.
The 32-year-old said: "This means a lot to me. Firstly, I'm joining one of the best clubs in England.
"When I left City it was important for me to stay in England because it's the best league - and to join Liverpool is massive for me.
"I chose Liverpool because of the history and because of the spirit of the team and the people here."