The council plans to leave its Abbey Foregate base in Shrewsbury, and intends to build a new Shirehall in the town's centre as part of proposals to redevelop the Riverside area.
Campaigners from Save Our Shirehall have repeatedly urged the authority to reconsider, and turned out to protest at a council meeting earlier this week.
Now Shrewsbury Civic Society has added its voice to the debate over the building's future, calling for guarantees it will not be demolished.
The council previously fought successfully for a 'certificate of immunity' for the building.
The certificate prevents the building being granted listed status, making it easier for any future owner to demolish.
Speaking at the meeting of full council, Councillor Dean Carroll, the cabinet member for physical infrastructure, pulled no punches, saying: "The building is broken, technically, functionally, economically and in terms of its green credentials."
But, in a letter to Shropshire Council's leader, Councillor Lezley Picton, the chairman of Shrewsbury Civic Society, Michael Dinneen, said the building should be protected.
He argued that although it may not be currently appreciated for its architectural merit, it is an 'iconic' Shropshire building and could be appreciated by future generations.
He said: "Shrewsbury Civic Society (SCS) is very concerned that there may be an attempt to demolish the Shirehall.
"SCS is opposed to demolition and seeks a clear statement from Shropshire Council that it will not try to demolish the building or sell it to a buyer or subsequent buyer, without ensuring that there are legal covenants in place, in perpetuity, preventing such a demolition.
"The debate over whether Shropshire Council continues to occupy the building is not an issue for SCS.
"Our concern is to make sure that the building is not ever demolished.
"The Shirehall is an iconic Shropshire building. Its modern style of architecture is not appreciated by everyone at this time, but that does not diminish the standing of the building and its surroundings.
"The Shirehall is appreciated and recognised by architectural critics such as Sir Nikolaus Pevsner and Owen Hatherley, the 20th Century Society, Historic England, SAVE Britain’s Heritage and other lovers of exceptional architecture.
"Demolition would deny the possibility of that admiration continuing and expanding, as the years unfold, and as the unfashionable becomes fashionable as has always happened with previous styles of architecture."
He added: "The Shirehall represents more than any other building Shropshire’s progressive contribution to the built environment in the 1960s.
"By ensuring the survival of the Shirehall, Shropshire Council can seize the chance to show how it can take a lead in conservation matters and progress the fight against global warming whilst saving an architecturally important building that cannot be replaced."
Shropshire Council has been contacted for comment.