It follows as a planning application was put forward by Ms Seema Jand for the site on Bridge Street.
The building was previously used by Nottingham Building Society, which closed down at the end of 2022.
The proposal involves converting and extending the first floor of the property, along with building a side extension.
Each flat will include a bedroom, kitchen and dining room, with the ground floor to remain unchanged and let out as a commercial space.
In a planning statement published, it was said that the conversion will introduce a “high quality accommodation unit” in a “sustainable location” of Fakenham.
The statement added that the proposal does not result in a “detrimental external impact” on the building or the wider street scene.
A decision on the application is due to be made by North Norfolk District Council at a later date.
To view the plans in full, visit the district council’s planning portal, using the reference: PF/25/2307.