Carragher, who has notched up 719 appearances for Liverpool since making his senior debut back in 1996, is out of contract at the end of the season and has hinted he could hang up his boots.
The 34-year-old has played a bit-part role on Merseyside under Rodgers and Kenny Dalglish before him, with Daniel Agger and Martin Skrtel preferred in central defence.
But Rodgers is keen to see him stay on after his deal expires and feels he still has plenty to offer.
The manager said: "He and I have had loose discussions in terms of where he is at. 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.
"It is different for defenders as well. 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."
Sturridge
Meanwhile, Rodgers has hit back at talk that new signing Daniel Sturridge, who arrived at Liverpool from Chelsea this week, has only moved to Anfield for the money.
He added: "If it was about money, he wouldn't be here.
"I've seen reports that he came here for £80,000 a week, which is absolutely ridiculous. It's nowhere near that.
"He knew this was a football decision and he knows that what's important is being happy."
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