The Hoops are currently sixth in the Scottish Premiership table, having lost three games in their first six outings this season.
Celtic are rebuilding under new boss Postecoglou after a season which saw rivals Rangers ending their hopes of clinching a tenth top flight title in a row.
With Celtic struggling for consistency at the moment, there have been suggestions the club are in disarray, but ex-Hoops boss Barnes has rubbished such claims and stressed they are still adapting to playing under a new boss.
However, Barnes feels Rangers have dropped their level compared to last season, while Celtic now have the added impetus of trying to get back to the top of Scottish football.
“Had Celtic won the Old Firm Derby; they would currently be top of the Scottish Premiership”, Barnes told Bonus Code Bets.
“They also lost in the Europa League but it’s not as doom as gloom as people are trying to make out. Rangers also drew on the weekend.
“Celtic are doing okay despite some people saying that they are in disarray, which they’re not.
“They lost a game which they’re allowed to.
“It’s only been six games in the league and they’re still getting used to the new manager, so they’ll be fine.
“Celtic will push Rangers closer this season because Rangers have dropped their level a bit.
“After stopping Celtic making 10 in a row, they’re not playing with the same determination and intensity that they were doing.
“Celtic have a new manager and new players have the added impetus of trying to get back to where they were.”
Hoops fans will be hoping Postecoglou will succeed in getting the best out of his men on a consistent basis soon as they look to knock Rangers off their perch this season.