The landmark building at the top of Park Street on Queens Road, was built in 1905.
Housing around 2.5 million objects, it attracts more than 300,000 visitors a year.
But several parts of the building are ageing and have “major defects”, according to the Local Democracy Reporting Service, external (LDRS).
A committee report said a recent survey showed the Edwardian Baroque building was in a “poor condition” and its exterior had “major defects”.
It said protecting the “fabric and longevity” of the building would allow the museum to continue as a “community resource and civic focal point”.
“We aim to make our building more accessible for everyone and engage the community in preserving this listed building and reinterpreting and accessing collections,” it said.
“This is a major opportunity to secure significant external funding that is highly unlikely to come from within the council given current financial pressures”.
The council has already received £680,000 in funding from Arts Council England for previous works, which were completed in March.
But another more recent bid for funding was unsuccessful.