Carroll's future has been under speculation since Rodgers arrived at Anfield this summer, with Newcastle United and West Ham United both linked with a move for the target man.
The 23-year-old came on as a substitute to score for Liverpool in Sunday's 3-1 friendly win against Bayer Leverkusen, an impact which impressed Rodgers.
"It was a terrific goal from him. It was a nice, clever piece of play by Charlie Adam as the ball came into him and he flicked it around the corner.
"Maybe eight or nine times out of 10 he would have expected the ball back, but big Andy showed his skill and ability as well.
"He's got stick that he can't play this way, but it was a wonderful piece of skill letting the ball go through his legs, getting the shot on target and tucking the ball away very well. It was very pleasing all round."