Gerrard has ended months of transfer speculation by committing his future to the club.
The signing of Cole appears to have been key to the midfielder's decision, with Gerrard saying: "It is a great signing for us. I know Joe well through the England set-up and have seen first hand just how good a player he is.
"He has proved his ability over many years in the Premier League - sometimes against us - so it will be fantastic to play alongside him in a red shirt for Liverpool.
"I wanted the chance to meet Roy Hodgson privately and having done so, I'm very impressed with his plans for the future."