McLeish, who won league titles, Scottish Cups and League Cups during a glittering Aberdeen career, hopes more than anything that the Dons can take Celtic all the way this season.
But he believes that they will need a lot of luck with injuries and suspensions - and that is just unlikely to happen.
He said: "If Derek McInnes keeps everyone fit then you have to think they might have an outside chance but you have to think Celtic will regroup and come again.
"It is healthy for the game that we have a level of competition that we haven’t seen for so long but ultimately it is difficult to envisage any team pushing Celtic all the way. Celtic will regroup in January and come again.
"You can’t ask any more of Aberdeen,” McLeish added. “They are mentally strong as a team. Derek has given his side all the evidence they need to believe they are a good team.
"But the challenge comes in sustaining it over the course of a season when injuries and suspensions come around. These factors play such a significant part when you have a relatively small squad and there is no doubt Celtic have far more depth."
Source: Sunday Herald