Planning documents show the two-storey premises is currently accessed through doors onto Howard Road, the Local Democracy Reporting Service said.
In the plans, one flat would be accessed through the existing entrance, while a second flat would be accessed via a new door created on Cromwell Street.
Meanwhile, flats three and four will be accessed through a reconfigured door on Howard Road with a lobby and new stairwell from the ground to first floor.
A stairwell would be installed allowing access into the former bank vault, after the venue was converted to a bakery from a branch of Yorkshire Bank.
The site is located within an established residential area and is flanked by terraced dwellings on both sides, documents added.
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