The Post Office is proposing to shut the branch on Cleveland Street and replace it with counters based in the WHSmith shop on New Market Street.
Under the proposals, there would be four Post Office serving positions in WHSmith - three open-plan counters and one screened - along with two self-service kiosks.
The staff currently working on Cleveland Street will move to the new site and the same services will be on offer.
It is proposed the branch would move in August.
The public consultation, which has been open for six weeks, allows people to have their say on factors such as the location, the proposed premises, accessibility or community issues that they would like considered before a final decision is made.
The consultation closes on Wednesday, June 15.
People can get involved by visiting postofficeviews.co.uk.