The Scot is finally set to resolve the ongoing saga by penning a new deal this week.
McLeish's profile has risen after he guided Birmingham to a ninth-place finish on their return to the Premier League last season and his delay in committing himself to the Blues has led to speculation about his future.
Jerome told the Birmingham Mail: "We want him to stay and I think he will stay. He has done a magnificent job since he has been here.
"The way he has brought Birmingham on, on a shoestring budget really, and pushed us into mid-table.
"He is an up-and-coming manager. That's why the likes of Liverpool and other teams will come knocking when they are looking for a new man. He has proven his worth."