Glasgow City Council, which owns the B-listed McLellan Galleries in the city centre, is marketing the building for clients who can run it on a long-term lease.
The galleries, which lay in disuse for many years but have recently been used for the Glasgow International Festival of Contemporary Art and the Royal Glasgow Institute of Fine Arts.
The building, at 270 Sauchiehall Street, is not for sale and has been a key feature of the city's art scene since the 1850s.
Interested parties have until noon on October 8 to submit a business case to Glasgow City Council.
A recent feasibility study indicated strong market demand for the building.
The council is looking for a tenant who would take a long-term lease on the galleries, with a "commitment to repair the building and develop it in line with its historic use, focused on art, culture and public access."
The bidding exercise will consist of two stages, with a business case and a formal presentation "concentrating on the community benefits of each bid and highlighting the relevant experience of each party in delivering art, culture, public access and community benefit in other projects."
Bailie Liz Cameron, executive member for jobs and the economy at Glasgow City Council, said: "The McLellan Galleries is a fantastic resource for the city, so it is great to see this process moving on to the next stage.
"The building has a great tradition, and we very much want the new tenants to continue to provide the services that have served the city so well in the past.
"I look forward to a number of very strong proposals being put forward for the long-term use of the Galleries."
An Open Day will be held at the Galleries in early September for interested parties.
It is hoped that a report recommending a preferred bidder will come in front of the council's Executive committee before the end of 2014.