Carroll joined the Reds in a big-money deal from Newcastle 18 months ago, but he is not part of Rodgers' future plans with the manager admitting he will only be used as a 'cover player'.
The England striker appears keen to stay and fight for a place, despite interest from West Ham, but Rodgers has suggested he should seek regular first-team football elsewhere.
"You don't need to be a rocket scientist to see Andy's been a cover player but he's been excellent in his attitude and acceptance of where he is at," he said. "But, as a club, I'm not sure we are in a position to have £35million players as third-choice strikers, or a winger on £5-6m a year.
"There's still a wee bit of thinking time, and it's hard to walk away from here.
The ones who want to play will probably look elsewhere, irrespective of finance, because that's their passion."