McLeish managed Coyle at Motherwell and the Clarets boss said: "Alex is a personal friend of mine and I have got a lot to thank him for in my career. He's doing a fantastic job at Birmingham.
"When you think of all the talk of takeovers and everything else, he's not let that affect anybody, he's got them playing some terrific football this season and there is no doubt that they have settled into Premier League life.
"Birmingham have a fantastic manager and he'll do his utmost to keep them up. I'll concern myself with Burnley, but I'm always looking out for my friends and how they are doing and I have no doubt that Alex will do fantastically at Birmingham."