"Obviously it's been a long time since we won the league and when I talk to fans in the street, that's the one that they'd dearly love to see at Anfield in May," said the defender.
"I know where they are coming from and I'd love to be a league winner, so we're going to give it everything this year to win our first Premier League title.
"The board have backed the manager off the pitch with his signings and now it's down to us to back him on it."
Carragher, who confirmed he will be back from his broken rib for Liverpool's next game at Portsmouth on September 15, dreams of seeing Steven Gerrard hold the Premier League aloft at Anfield before they move to their new stadium in 2010.
"We've got the new stadium to move into in Stanley Park in a couple of years' time and that's going to be a real buzz to be part of," he said.
"It would be nice to be able to bring the title back here to Anfield at least once more before we leave.
"It's the place where we brought the first one to and it would be fitting if Stevie held it aloft at Anfield before we go."