Cole made his first start for the Reds in over three months in the 3-1 defeat at Anfield, but was taken off at half time in a disappointing night for the former Chelsea man.
Rodgers told reporters after the game that the Cole saga left him particularly frustrated by the amount of money the club has spent on him, with his wages thought to be over £100,000 a week.
Cole was signed two summers ago under Roy Hodgson and the transfer was complete by the then Managing Director, Christian Purslow.
“The club have invested an astronomical amount of money on a talented player and Joe had the opportunity,” said Rodgers.
“He has been back fit a couple of weeks and his opportunities have been limited but you have to see.
“I thought it was difficult for him, it was too slow and it wasn’t what I would expect from a team I tried to set up to be dynamic.”