It argues the structure should be removed as part of the commemorations for the centenary of the first Austrian Republic, as well as the 80th anniversary of the "Anschluss", or annexation of Austria by Nazi Germany.
Memory Gaps has also suggested that prior to it being dismantled, a "speech for peace" could be given from the balcony on November 12 - a hundred years to the day since the Republic was proclaimed.
The city authorities seem to have been caught off guard by the proposal.
Vienna's top cultural official, Veronica Kaup-Hasler, said she welcomed the debate and the fact that it had drawn attention to a forgotten detail of the town hall's history, a spokesman told the APA agency.
But Kaup-Hasler preferred that the balcony remain in place accompanied by a clearer explanation of its history.
The head of the commission responsible for researching into and returning property stolen by the Nazis, Eva Blimlinger, agreed, telling the Kurier newspaper: "This balcony is, like so much that resulted from national socialism, a part of our history."
The Anschluss of 1938 was welcomed by a broad swathe of the Austrian public.
Shortly afterwards Hitler held a famous speech from another balcony, of Vienna's Hofburg imperial palace on March 15.
That speech, and the one in front of Vienna's town hall, drew massive crowds.
Austria's relationship with its Nazi past remains a difficult issue even today, not least since the far-right Freedom Party (FPOe) entered into a coalition government last year.
Earlier this month, Vice-Chancellor and FPOe leader Heinz-Christian Strache sparked controversy when he unveiled a monument to women who cleared debris from the streets during the war, with some accusing of him of seeking to rehabilitate the reputation of some Nazi-sympathising women.