Cole is looking for a fresh start and hopes to be part of a new era under Rodgers after failing to impress since his summer move to Anfield two years ago.
The former Chelsea player wants to make a success of his Liverpool career, and feels that he has something to prove to the fans.
"It's great to be back," Cole told the club's official website. "It's a clean slate for everyone, so I'm delighted to be back. Training has been good - and it's nice to be back in England and nice to be back in Liverpool.
"I don't want to leave Liverpool one day having not been a success here or having left something here. I want to make an imprint at the club.
"I've been a success at West Ham, Chelsea and Lille - I don't want people to be able to say that I haven't here.
|BACK IN THE BIG TIME
|2/1||Liverpool are 2/1 with Bet365 to finish in the Premier League's top four next season|
The 30-year-old spoke of his reasons for joining the club and insists that he is still passionate about playing for the Reds.
He added: "If I get given a chance to play, I'll always give my best and give everything I've got to the team.
"I came here for one reason - because I wanted to be a part of the resurgence of this club.
"It excited me coming here because I knew what a big club it was with a rich history and I wanted to be part of something. I still do.
"The fire still burns in me to be a part of it - I want to be part of this club moving in the right direction."