Carragher was speaking after their defeat at Everton.
He told Sky Sports: "You just run for this man, in these last four games. No-one will say the title's gone, but we all know the teams they're competing against.
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"This guy, and what he's done for so many careers in that squad, you run like your life depends on it. That's what you've got to do, give this man a send-off.
"Whatever happens, his legacy's there. He's an absolute legend, he's loved by every Liverpool supporter. The players there might think the title's gone - they won't say that to each other - but just run your heart out, not just for the club but for the manager, between now and the end of the season."