The two buildings, built in 1913, were first used as a military clothing factory, and later as university student accommodation, reports the BBC.
They were also used as a makeshift hospital after the Second World War bombing.
The bombing on August 6, 1945 flattened most of the city, and as of last year only 85 buildings built before the bomb remained within 5 km of "ground zero".
The buildings survived because they were made from reinforced concrete with a red brick exterior. Some bomb damage to the metal windows and doors is still visible.
In 2017, authorities found the structures - now publicly-owned - were highly likely to collapse in a strong earthquake.
As the buildings are not in use and not open to the public, the local government decided they should be demolished by 2022.
A third building at the site will be preserved, and its walls and roof will be repaired and reinforced to protect it from earthquakes.
Iwao Nakanishi, 89, was in one of the buildings when the city was bombed. He is now the head of a local group demanding their preservation of the buildings.
"Considering the historical significance of telling the tragedy to the future generation, we can no way accept the demolition," th BBC quoted Nakanishi as saying to the Japanese Mainichi newspaper.
"We strongly oppose it."
Nakanishi said the buildings could be used to promoting "the abolition of nuclear weapons".
As a direct result of the bomb, around 80,000 people were killed and 35,000 others injured.
( With inputs from IANS )