The Northern Irishman has stated that the club is not in a position to retain "third-choice strikers" at such a cost but praised the forward's attitude.
Carroll has been linked with a return to his former club, Newcastle United, and was also the subject of a loan bid from West Ham with a view to a permanent transfer if the London club achieved Premier League survival.
The 23-year-old is the Reds' record signing by some distance but has found his opportunities at the club limited since Rodgers arrived from Swansea City to replace Kenny Dalglish as manager.
"You don't need to be a rocket scientist to see that Andy has been a cover player for us," Rodgers told reporters.
"He has been excellent in terms of his attitude and his acceptance of where he is at but, for us as a football club, and I am talking generically here, I'm not sure we are in a position to have 35m pound players as third-choice strikers or wingers who might be on 5m-6m pounds a year. I don't think the football club is in that position.
"This is the challenge that I have to work something but it is not going to be done overnight."
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