Cole joined Liverpool on a free transfer following the expiry of his contract at Chelsea, and Gerrard believes he is one of the most talented players in world football.
"Messi can do some amazing things, but anything he can do Joe can do as well, if not better," Gerrard told Liverpool's official website.
"He used to shock us in training by doing footy tricks with a golf ball that most players can't even do with a football. I really fancy Joe for the [Player of the Season] award this season."
Gerrard also said he believes Liverpool will force their way back into the top four this season.
"This is a great club and after speaking with Roy Hodgson I'm really pleased to see his plans for the club," he said. "It's a new era now - we must forget about the horror of what happened last season and get this club back to where it belongs, which is the Champions League at the very least."