The Reds deemed the England international surplus to requirements last summer, allowing him to link up with Premier League rivals West Ham United on a season-long loan.
The Hammers have expressed a desire to make that move permanent, with Carroll having offered enough during an injury-hit campaign to suggest that there is more to come from him.
Rodgers agrees that the 24-year-old has the ability to become a star turn for club and country, but he is not entirely convinced that Anfield is the right stage for the burly frontman.
On a man Liverpool paid £35million for in January 2011, the Reds boss said: "It was very unfortunate for Andy when he joined this club because he had limited experience of the Premier League.
"Through no fault of his own he arrived on the back of an incredible price tag for someone so young and inexperienced, and the early stages were difficult for him.
"But there's no doubt Andy has great qualities. Look at one of his two goals last week for West Ham from the corner (in a 3-1 win over Wigan).
"There are not too many better sights in football when you've the big guy going and heading the ball like that.
"As to his future, there are other things involved, it's not as simple as him being here. There are other things we need to consider in it all.
"But do I think he can be a top player? Is he a top player? He certainly has that potential. If he gets that opportunity of playing in a team that suits his style, then there's always a chance."
A return to Newcastle United has been touted, with it speculated that the Magpies would be prepared to sanction a swap deal involving French winger Hatem Ben Arfa in order to bring Carroll back to Tyneside.
On those rumours, Rodgers added: "That's normal, that's the football world. There's always gossip and speculation.
"The important thing is Andy is at a very good club with a very good manager.
"As I've said, it's one where we will see it through to the end of the season and we'll take it from there."