The former Reds forward also believes that Brendan Rodgers is the right man for the job at this moment in time.
They also required penalties to see off League Two side Carlisle in the Capital One Cup in midweek, in what was another disappointing display that has increased the pressure on Brendan Rodgers and his side.
Finishing in the top four seems an unlikely prospect at this moment in time, but Barnes has backed the Reds to push hard for Champions League qualification.
“I believe Liverpool will get it right to challenge for fourth,” he told TalkSport.
“I understand the expectations and, as to whether they can come back and win the league or finish second, probably not, but I still believe if people stick with the club then we can challenge for fourth.”
Rodgers the right man for this team and squad
The former forward has also backed Rodgers, despite the increased speculation about his future at the club.
With his side once again struggling at the beginning of the season, as they did last year, rumours that Rodgers will lose his job have been abound, with Jurgen Klopp and Carlo Ancelotti both being touted as possible replacements - by the Metro, among others.
But Barnes believes that Rodgers is the right man for the squad at Anfield as it stands, suggesting that a change at the end of the season, or in January would make more sense.
“Brendan is the right man for this team, this squad, as it stands,” he added.
“If you are talking about bringing someone in in January or the end of the season and spending lots of money, maybe then you can start talking about changing the manager.”