The striker, signed for £35m from Newcastle United in January 2011, has been subject of transfer speculation since the arrival of Rodgers as Reds manager in June.
"It's still too early to say," said Rodgers, when asked if Carroll was staying at Liverpool.
"You don't have to be a rocket scientist to see Andy has been a cover player for us in the opening few games. His attitude has been excellent as has his acceptance of where he's at.
"But for us as a football club, and I'm talking generically here, I'm not sure we are in a position to have £35m strikers as third-choice strikers, or wingers who might be on £5m-6m a year."