Carragher accepts the players have shown a marked improvement under new manager Kenny Dalglish and hopes to see the Kop idol handed the job on a full-time basis.
"We've obviously improved," he told liverpoolfc.tv.
"I have great respect for Roy (Hodgson) so I would never say that things have changed for the better and all that type of stuff.
"As a club and as players we didn't do enough to help the previous manager. Our performances weren't good enough and the manager took the brunt of it. Obviously things have improved now and I think Kenny coming in has got the supporters and everyone onside.
"It's not our decision (if Dalglish is appointed full-time) - it's up to the owners. He's a hero to me and Stevie and the results are going well. That stakes his claim and if you are asking me, then I'd love him to be the manager (permanently)."