The former West Ham and Chelsea midfielder was farmed out on loan to French club Lille this season, scoring nine goals in 42 appearances in all competitions.
But Cole believes that he has unfinished business at Anfield and “would love” the opportunity to rectify his reputation after spending only one season playing for the Reds.
Speaking to Sky Sports, Cole said: "I've been a success at West Ham, a success at Chelsea.
"I've moved to another country in France and had a good season.
"It's just that season at Liverpool - I would love to have a chance to put it right, if that's what they wanted."
Lille are keen to keep Cole at the club next season, especially with the much-coveted Eden Hazard set to move to Chelsea this summer.