McLeish guided Blues to ninth spot in their first season back in the top flight and knows expectations are higher ahead of the opening game at Sunderland.
But the Blues 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."
Source: PA
Source: PA