Chile winger Jean Beausejour, Czech Republic defender Martin Jiranek and Belarus midfielder Alexander Hleb joined Blues on a busy transfer deadline day, prompting talk the club could improve on last season's ninth-place finish.
But McLeish does not want fans to get carried away and insists he will be content first and foremost just to stay in the Premier League.
"That's nonsense to start talking about that," he said. "I thought silly season finished when the window closed.
"I don't think we should be thinking about that. We're still a fledgling Premier League team. We want to establish ourselves and develop again.
"If we finish in the top 17, like the targets I set last season, then I'll be happy.
"Some people might not be happy with that but for me there's around eight teams that are going to be pretty hard to keep in touch with over the course of the season.
"There's another 12 out there playing to stay in the Premier League."
McLeish is confident, however, that his new arrivals will move Birmingham in the right direction.
"Collectively they'll enhance the quality of the squad," he said.
"We try to take a step up with each season, and sometimes you take a step up within each window.
"We believe that we have again enhanced the overall quality of the squad.
"But it doesn't guarantee you anything because we know that we're not a surprise package this year and that teams will be waiting for us.
"We're well aware of that challenge, and I'm throwing the gauntlet down to the players that did so well last year, and saying to them 'do it again guys'."
Jiranek has loftier ambitions than his new managers, though, and reckons Europe is a distinct aim.
Speaking through an interpreter, he said: "Is it seven teams that go into Europe? Then yes, I think so."