The 30-year-old midfielder endured a poor campaign with the Reds after joining on a free transfer from Chelsea in July 2010.
He resurrected his career in 2011-12 by helping LOSC finish third in Ligue 1, scoring nine goals in 42 appearances for Rudi Garcia's side, but admits the "fire still burns" to prove himself on Merseyside.
"I would love the chance - if they wanted to - to put it right," he told Sky Sports News.
"I don't want that on my career. I've been a success at West Ham, a success at Chelsea, and come to another country in France and had a good season.
"It's just that season at Liverpool. It's in their hands. I'm easy whatever way but the fire still burns with me and I would love to have the chance put it right if that's what they wanted."
And Cole believes the dismissal of manager Kenny Dalglish earlier this month may open the door for a return to Anfield.
"Kenny was a good guy - he didn't lie to me. I wasn't in his plans," the player added.
"It [Dalglish's departure] just changes my future. Whereas it was quite clear before Kenny got the sack that I wouldn't be going back there, who knows what will happen now? I'm just waiting on some kind of information."