The 23-year-old striker, who is on a season-long loan at West Ham United from the Merseyside club, was bought by former Reds boss Kenny Dalglish for £35m but was later offloaded when Rodgers was appointed manager.
"I haven't really had a fair chance at Liverpool," Carroll told ESPN. "I was really told that I should move on to keep playing regular football. Obviously, I didn't really get a chance there.
"A new manager came in and I wasn't part of his plans. For me to play football, that was my chance to do that. That is why I am here at Upton Park.
"I think I could fit into every system. I believe in what I can do. Obviously [Rodgers] has his beliefs and he didn't think that. But for me, obviously, it was better to move on instead of waiting for my chance, which he told me might not come."