The only question Rodgers needs to ask is "will Andy Carroll be able to fit into our style of play and philosophy?" and he clearly has his doubts.
When you play the way Rodgers wants us to you need a striker who is mobile, energetic and willing to make dozens of runs which may not result in him getting the ball, but will pull defenders about and open space for others. He needs to be able to have a turn of pace, good technical ability and be able to chase lost causes. To me, that doesn't get the best out of Andy Carroll as he wants the ball into his feet or head and then bring people into play - in essence a classic target man who stays central and suits a direct game.
I don't find it surprising that we may consider letting him go and it would possibly best for Carroll himself to go out and get first team football elsewhere. No point keeping him on the bench and letting him get frustrated, and if he is out playing for a team who gets the best out of him he will retain or even increase his value.