McLeish is confident Dunne will be available for the final two games of the campaign against Tottenham and Norwich. It is also a boost for the Republic of Ireland ahead of their Euro 2012 campaign.
Dunne has been sidelined since suffering the setback against Manchester City in February.
McLeish said: "It's good news regarding Richard. He has rejoined the group for training today (Monday). If he doesn't make the game at West Brom this weekend, he should be ready for the Tottenham match."
McLeish expects Dunne to be available for instant first-team action rather than having to test out his fitness via the reserves.
He said: "With a guy like Dunnie, and with what the modern-day physical trainers and medical people do, the aim is to try and get the player up to speed so he can go straight back into the side. There should be plenty of preparation work for Dunnie so he can step straight back in."