"I don’t know what Damien said and it’s none of my business," added the Reds boss, who was pleased to see Cole score in Sunday’s 3-2 win at West Ham.
"It was important for Joe to contribute. This season has been very difficult for him because he has been injured for the majority of it.
"The responsibility has always been with the players and if Joe continues to work how he has been when he gets the opportunity he has to take it.
"He works very hard at his game and when the chances come I know in his own mind he wants to take them and be effective for the team.
"That is the challenge for all the players. It is a challenge for us all to improve and we want to keep rising up the table."
Top scorer Luis Suarez returns after a one-match suspension for the visit of Aston Villa tomorrow and although his goalscoring exploits have been impressive the Uruguay international still has to carry the stigma of being labelled a diver.
However, Rodgers believes recent incidents involving other players has highlighted that there is a wider issue to be addressed.
"Luis Suarez was held up as the only one that seemed to be doing it," he said.
"Whenever many other players were doing it, both British and European, it seemed to go very quiet.
"It’s something that is the responsibility of the players. The managers can only guide and give their opinion to the players.
"You see it now becoming a part of the game but it’s certainly something we want to eradicate.
"It’s difficult for the referees because they are having to make that judgement straight away.
"Pundits and others get a chance to review it a number of times."
Rodgers also said the club had learned their lesson from last summer’s transfer window when their pursuit of Clint Dempsey prompted a complaint from Fulham, which was only withdrawn following a letter of apology from Reds chairman Tom Werner.
"There is nothing for me from a managerial perspective to add to it," said the manager.
"As a club we’ve all moved on from the summer and when January comes whatever experiences we had then we’ll be better for."