The building is Grade II listed and in need of major repair and refurbishment.
Developers interested in bidding for the 50-year-old building are being invited to have a look around and consider new ideas, which could see the 15-storey tower used for office space, commercial leisure use such as a hotel or residential accommodation.
A contract notice has been published in the Official Journal of the European Union advertising the Civic Centre as a £20m to £35m redevelopment opportunity.
The council described its approach as a formal procurement process designed to encourage bidders to discuss their ideas in detail with the council.
The development opportunity covers the high rise block, the two-storey podium, single-storey flying office accommodation and the rear ground level public car park. The Council House and car park are not part of the proposals.
"The council has been wrestling with this issue for some time now and time is ticking on,” said councillor Mark Lowry, cabinet member for finance. “We’ve brought this project forward and hope that by talking through options in confidence with developers we will get a clear sense of direction.”
The council hopes to find a buyer for the building, but options may include leasing part of the office space back or leasing other office accommodation in the city centre.