Dunelm House was built in the 1960s and although fans believe it to be a prime example of the architecture, it has not been given listed status.
The university said the concrete structure needed repairs costing Â£14.7m and wants to demolish it.
A petition calling for it to be refurbished and maintained has attracted hundreds of signatures.
The building splits opinion in the city, with some residents referring to it as a "concrete carbuncle".
However, architect James Perry, from the Something Concrete and Modern project, which is documenting post-war buildings, described it as "brilliant" and "incredibly polite for a modernist building".
He said: "It is expensive to demolish and dispose of a building.
"I would say as an architect we need to be looking to to reuse, recycle, extend or adapt our buildings rather than disposing of them because we don't like the way they look."
Durham University said that if demolition went ahead then the replacement building would be of high quality and a world class design for such a sensitive location.
Campaigners have called for Durham University's "brutalist" student union building to be saved from demolition.