McLeish led the Blues to an impressive ninth-place finish in the Premier League last term just 12 months after they were promoted to the top flight as Championship runners-up.
Several teams in recent years have enjoyed initial success after coming up from the second tier only to struggle in the campaign that followed, but Coyle - whose side host Birmingham on Sunday - believes the Midlands outfit will build this season.
"Alex is a top-class manager and he's been able to improve the squad from last year with signings," Coyle said.
"Joe Hart went back, but he has got a terrific replacement in Ben Foster and Nikola Zigic has come in - they have brought some very good players to the football club, and spent some money doing it.
"I have got no doubts that Birmingham under Alex will carry on the good work from last season.
"I think particularly under Alex's stewardship, they will go from strength to strength.
"They are a big club and he is doing the right things to take them on."
Both Bolton and Birmingham have made encouraging starts to the 2010-11 campaign, with four points apiece from their opening two games.
Wanderers have been tipped in some quarters for a top 10 finish themselves, but Coyle has stressed that it is still very early in a long season.
"Everybody wants points on the board and four points from two games has been a pleasing start, but that is all it is," Coyle said.
"We have to concentrate for the Birmingham game, and try and do our utmost to win against a very good team and a terrific manager.
"He is a good friend of mine, but Alex knows and I know that during the game, we will be going all out to win the match for our respective clubs. We have to continue the hard work we have already done."
Asked if he felt Bolton could potentially target what Birmingham achieved last term, Coyle said: "Birmingham had a terrific season and enormous credit must be paid to them for that.
"There are clubs we would probably bracket as being in our area, and it shows that it can be achievable, to try and push yourself into that top 10.
"But that doesn't take two or three games - that takes a whole campaign, 38 games, and come the end of that, we'll see where we are."
Coyle and McLeish worked together in Scotland and the Bolton manager hopes there will be time for a catch-up once the match at the Reebok Stadium has finished.
"He was my manager - he signed me from Dundee United and I played under him at Motherwell," Coyle said.
"I loved every minute of it and have a terrific rapport with him. He has helped me along the way, there is no doubt about that, and hopefully I helped him when I was a player for him.
"I am very respectful of the job he has done everywhere he has been - he has been different class.
"I'm pretty sure after the game we'll sit down and have a chat, but during it, we'll be going all out to win."