The Reds manager believes that the 34-year-old defender has more to offer at Anfield and hopes to discuss fresh terms before the end of his current deal, which is this summer.
The former England international has hinted that he may retire in the summer, but Rodgers believes the 34-year-old has more to offer despite struggling to displace Daniel Agger and Martin Skrtel from the first team. Carragher has played nine matches in the Premier League this season, but eight of those appearances have come as a substitute.
"He and I have had loose discussions in terms of where he is at," Rodgers told reporters. "We spoke before Christmas and decided we'd have a chat after Christmas. We had a busy period coming up, but there is no doubt I want him to stay as a player because he still has a lot to offer."
Carragher made his debut for Liverpool in 1996 and has gone on to make 719 appearances for his boyhood club, but Rodgers believes he is still in good enough condition to play in the top flight.
"It is different for defenders as well," Rodgers said. "The older you get, you are in that firing line at the back and there is no hiding place. He has been a pivotal player, a real iconic player for the club - he is still in great condition, he looks after himself.
"He turns out every single day and I'm sure there are days that are difficult for him as a player when you don't play. But he is a wonderful example to any player, absolutely incredible. And he has still got a few years left in him yet, for sure."