The 30-year-old has scored four goals in 27 appearances for Lille, including the second in his side’s recent 4-1 victory over Ajaccio.
Cole joined Liverpool on a free transfer from Chelsea in July 2010 but the England midfielder struggled to make an impact at Anfield.
“It’s something I would like to do [extend stay at Lille],” said Cole.
“It’s not in my hands at the moment but I’m very happy here, my family is very happy here.
“We have to wait to see where Lille finish in the league and then start planning for next season.”
Cole, who has 56 England caps, has not featured for the national side since their disappointing World Cup campaign in 2010 but remains positive of a call-up to Euro 2012.
“I think it depends on who the manger is,” added Cole. “It’s something I would love to do.
“I love playing for my country and if it happens I’d be very happy to play against France in my first game.”
His team-mate Eden Hazard has been linked with a move to the Premier League this summer with Manchester United and Tottenham Hotspur reportedly interested, and Cole believes the French midfielder would be a success in England.
“He’s a special talent and he will be a success wherever he plays,” Cole said.
“I hope he stays here because it’s a pleasure to play with a player of such quality.”
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