Brendan Rodgers’ men have recovered from a sluggish start to the season and are now in 11th place, and Carragher reckons third and fourth place is up for grabs after Chelsea and Everton both lost on Saturday.
“We look as if we have not made a good start, but with winning on Saturday, we are not too far behind the top four,” Carragher told BBC Sport. “It is still up for grabs.
“Besides the top two, and even Manchester United lost on Saturday, a lot of teams are dropping points all over the place.
“From our own point of view, we now want to kick on and make a challenge for that top four place because it seems that everyone is throwing it away.”
The 34-year-old defender also praised Rodgers for placing his faith in club’s young stars, such as Raheem Sterling, after having limited funds available in the summer transfer window.
“The owners have come in and looked at it and thought they want to bring in a new manager who will promote players from the Academy, which he has done,” he added.
“I think it does take time to settle in. It was difficult to replace players like Maxi Rodriguez and Dirk Kuyt due to the financial situation in terms of us spending a lot of money in the last few years.
“[Rodgers] has been very brave putting young players in and giving them a chance. Hopefully, we’ll see the benefit of that in the future.”