The Gunners spent around £80m on new players over the summer, including a reported £30m on Alexis Sanchez from Barcelona.
Despite manager Arsene Wenger improving his squad ahead of the new campaign, Smith believes that Chelsea and Manchester City remain the top contenders for the league crown.
"Another trophy would be great this season, but I can't see them winning the title, I really can't," he told talkSPORT. "They're still a few furlongs behind Chelsea and Manchester City, as well as they might have done in the summer transfer window.
"Realistically you're looking at another top-four finish, and maybe they can nick a cup again, but unfortunately I still don't see them as title challengers."
Arsenal return to Premier League action on Saturday when they welcome champions Manchester City to the Emirates Stadium.