The Gunners are regarded by many as one of the clubs that will be challenging for the title, and their recent 3-0 win over Manchester City in the Community Shield only heightened expectations.
However, Carragher believes that Arsene Wenger needs to make vast improvements to his side if they are to win any more silverware over the next campaign.
"They had a good result against City in the Community Shield, but I look at the spine of the team and think - goalkeeper, good but not great, centre-backs, good but not great," he told the Daily Mail.
"[Mikel] Arteta, holding midfield player and captain, good but not great. Same with the strikers. The spine comes up short. They spent more time at the top than any other team last season but still came up short.
'Why has Arsene Wenger not done what [Jose] Mourinho has done? Why has he not bought a top goalkeeper? I wish Wenger would go out and buy a big goalkeeper, a strong centre-back, a big midfielder and a decent striker and go for it."
Arsenal last won the league title in the 2003-04 season.