Reports have suggested he could be set for a move but Rodgers said he will "see what happens" and insists the way the situation develops from here is very much down to Cole.
"I'm just going along game by game," said Rodgers. "There is every opportunity for every player. It's not just about the games. Every player gets a chance every single day to show what they are all about.
"Is his future here? You would have to ask him that. The guy is still a Liverpool player, and while he's a Liverpool player I'll work as hard as I can with my staff to try to make him the best we possibly can.
"If he's not going to play as often as he wants then he's no different to any player. Players are about game time and money.
"But he's not any kind of problem, he's a good guy and we will see what happens over the coming months."
Meanwhile, Fulham have received an apology from Liverpool chairman Tom Werner for his club's "unprofessional" behaviour in pursuing their former player Clint Dempsey during the off-season, prompting the London team to withdraw a complaint made to the Premier League.