McLeish guided Birmingham to ninth spot in their first season back in the top flight and knows expectations are higher ahead of the opening game at Sunderland on Saturday.
But the former Scotland boss has digested the pre-season thoughts of Pulis and claims consolidation and staying up are his main aims.
McLeish said: "Thinking of consolidation and just staying up sounds negative but we have got to be realistic at the club.
"We have to realise teams have come up before from the Championship and done well in their first season and then they've had a very difficult season.
"We are not a surprise to anyone any more. There is more pressure on us this season. Teams may tactically prepare a bit more thoroughly than they did last season.
"The players are always looking for improvement. Whether they can do better than last season, that's my challenge to them.
"But staying in the league is important and I keep bringing Tony Pulis' name up.
"He is a very wise man. He has been in the Premier League now coming up for his third year and said despite spending £8million (on Kenwyne Jones) and money like that, he is still talking about making sure they stay in the league.
"He said that would be the aim for three years in a row and he would then consider Stoke a fully-fledged Premier League team.
"With Tony saying things like that, despite the spending he has done over the last three years, which is more considerable than us, who am I to say 'we are going to finish eighth, seventh, or get into Europe?'
"We are not quite ready for that but we will keep trying to achieve it. Anything better than being a Premier League team next season is a bonus."
McLeish admits Blues are currently lacking in squad depth as he maintains his efforts to strengthen before the transfer window closes.
He said: "We lacked a bit of depth last year as well if key men get injured.
"We were lucky, touch wood, and hopefully we don't get injuries this season as well. Losing key men is vital to a club like ours.
"It was vital for a club like Arsenal when they lost Van Persie. They were going great guns and when he dropped out they started to have dodgy results.
"It stands to reason if we lose a key player it could be tough. That is why we are trying to make the squad stronger in depth."
McLeish knows the importance of Blues making a positive start to the campaign.
He said: "It is a tough away game, a tough start. We are at a loss as to why six of the last seven times in the Premier League, we've had our first fixture away from home.
"I can't understand that. If that is all about the luck of the computer, then we must be one unlucky team.
"But saying that, if we can get off to a flying start away from home, that will be fantastic for confidence.
"We've got to go up there and look to perform the way we did in the second half of the last game against Sunderland."
After getting off to a bad start and conceding two early goals Birmingham improved after the break.
"If we start the way we started that game, we won't win the game. We need to go up there and be positive."