Payet is now fully fit and hopes to challenge Cole for a place in Lille coach Rudi Garcia's plans.
"I don't mind the competition with Joe Cole," said the former St Etienne star. "I think its healthy for us.
"He has fit in well and is proving a decisive player for the team. Anyway, it's not just him. There's also Eden (Hazard), Ludo (Obraniak) and Moussa (Sow). Competition here is fierce."