Blues are facing competition from local rivals West Brom for the player but McLeish received a positive response when holding talks with Dembele's agent.
McLeish has said: "I had good vibes from the agent when I met him. Dembele is a quality player but I never like talking about potential signings until the deal is done."
Juventus midfielder Mauro Camoranesi also remains on McLeish's radar and is due to have talks in Birmingham on Monday.
The 33-year-old is available on a free transfer.
His agent, Vlado Borozan, said: "Let's just say that we will be in England to work out the final details and that we are already on the right path."
Source: PA
Source: PA