Carragher wants sustained Liverpool title challenge
The league title is the ultimate goal and it has continually escaped them since their last win in 1990.
"People always talk about us pushing for the league, but to be honest it isn't very often that we've even challenged," he told the club's official website.
"If we were challenging with 10 or 15 games to go then who knows?
"But just a challenge this season would be nice, to still be in the hunt.
"I think it's only happened once since I've been in the team, when I played a few games under Roy Evans.
"We probably should have won the league but let it slip towards the end.
"But since then we've never even challenged so that's something we're all looking for.
"I'd take a title medal over anything else at the moment because that would make my full set, but I just want to be competitive in the league."