The former Chelsea midfielder has made just four appearances in the Premier League and featured eight times for the Reds overall this season and the 31-year-old has claimed that he wants to "do well" for the Merseyside outfit.
"It's not just about the games," the Liverpool Echo quotes Rodgers as saying. "Every player gets a chance every single day to show what they are all about.
"Unfortunately for Joe since the [Europa League] game against Young Boys when he created a goal and scored a goal he has been disrupted. That's unfortunate for him because someone else comes in and takes the opportunity.
"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."
Cole spent last season on loan at Ligue 1 side Lille.