Andre Villas-Boas conceded Chelsea's hopes of challenging the two Manchester clubs - they trail them by 11 points - were remote after the 1-1 Boxing Day draw with Fulham at Stamford Bridge.
McLeish knows that will have hurt the Blues, but he has challenged his side to repeat their performances against Arsenal and Stoke.
The Scot also masterminded a single-goal win over Chelsea when Birmingham boss at St. Andrew's last season.
McLeish said: "It's a hard away game. Chelsea don't lose too many at their place. But they are starting to lose a bit of ground in the league as opposed to Manchester United and Manchester City.
"That will have hurt the Chelsea players and Villas-Boas and they will try to take that out on Villa to make sure they are still in the hunt.
"But we have got to have something to say about that."
McLeish added: "We have nothing to lose at the weekend. The players have got to throw the shackles off as they have done the last couple of games.
"We played some really good football in the Arsenal game and, if they go out with that same kind of attitude and outstanding performance levels, we can surprise people.
"We know that Chelsea will be favourites but sometimes the bookies don't get it right."
McLeish is looking for his side to start picking up positive results against the leading Premier League sides - something he achieved on a regular basis with Birmingham.
He said: "I'd like to see us doing more against the elite teams like Chelsea
"The win (for Birmingham over Chelsea) shows you it can happen and there were a number of good results against the big teams over the last two seasons.
"That is what I would like to start to do here but the performance levels are very important.
"That game last year saw Chelsea have the majority of the play but we managed to get one goal and defended well and sometimes you need a bit of luck."
McLeish admits he lacks the strength in depth to rotate his players and most of the squad in action at Stamford Bridge will also face Swansea at Villa Park on Monday.
He added: "We will go with the strongest team in both games. We don't have the capacity to rotate the squad.
"There were a lot of young, untried players sitting on the bench the other night (against Stoke).
"The same guys have got to go into battle again and we think their fitness is good and they will be buoyed by the confidence of their performances in the past two games.
"That helps to erase any tiredness."
Striker Darren Bent (thigh) and defender James Collins (Achilles) should be fit to face Chelsea.
Full-back Alan Hutton will also return after serving a one-game ban for his red card against Arsenal.
But veteran forward Emile Heskey is ruled out with his Achilles setback although McLeish is hopeful he will be available to face Swansea.