Carragher, 35, has announced plans to hang up his boots in June and has been linked with a job in the media, but his manager believes he could continue playing for the club next year.
The defender has made 33 appearances for Liverpool this season and has played the full 90 minutes in all but one of their last 11 Premier League fixtures.
Rodgers plans to use him in the first team for the rest of the season and says he will discuss the possibility of retaining his services at Anfield before his contract expires.
He told Sky Sports' Goals on Sunday: "He's been fantastic and I do [believe he could play next season]. Him and I communicate a lot and he's obviously got something in his mind.
"Him and I will sit down between now and the end of the season anyway and we'll see where it goes.
"If he does go, I will certainly miss him because he's been brilliant for me in my time there, going in as a manager of a club of that size.
"He's shown great respect and really helped me and the other guys. The thing for now is to keep performing the way he is until the end of the season and between then and now him and I will chat and we'll take it from there."
Carragher considered quitting Anfield last summer and Rodgers said one of his first tasks after taking the Liverpool job was to convince him to stay.
And while he was not initially planning to use him as a regular starter, he says the 35-year-old consistently impressed in training and soon forced his way into the team.
Rodgers added: "It's an interesting one with Carra because this guy's incredible. I want to put that on record.
"I saw him from the outside when I was at Chelsea and the games he was playing in the Champions League. I always thought he was one of the best defenders in European football at that stage.
"To come in and work with him at close hand and see his will and determination. His play speaks for itself, but the fight and spirit and quality that he shows every single day in training is amazing.
"I had a good conversation with Carra last summer because he was maybe thinking about moving on to pastures new last year, but we talked and decided that he would stay.
"For me, he's hardly put a foot wrong. I obviously wanted to go with a pairing of Martin Skrtel and Daniel Agger as maybe the future at the beginning of the season and those guys have done really well.
"But I felt that with Carra and all the games he's played - the cup games and any time he's come in in the league - he's been remarkable and I felt I wanted to put him in the team."