"Liverpool is already one of the most extraordinary cities in its subject matter and it is fitting that this museum is being created here.
"It is the legacy we will be bequeathing to future generations. Together we can make history."
Dr Fleming said the building would be "a great physical legacy of 2008" on a par with past achievements such as St George's hall and Liverpool cathedral.
The "warts and all" museum would reflect the port and people.
Dr Fleming, who said city museum and gallery attendances were growing faster than anywhere in the country, added: "The city needs a far larger museum to do it justice."