The bank, on Park Row in the city centre, is one of 32 across the UK being shut for good by its arch-owners Natwest.
The closures, which also include Natwest’s Headingley branch, were announced on Valentine’s Day this year by bank.
They did not reveal exactly when the banks will shut, saying only they would do so in the next 12 months.
But this week, Natwest has asked Leeds City Council for planning permission to strip out the cash machine at the RBS branch and tear down all signs relating to the bank outside and inside the building.
They have to formally ask for permission, as the premises is a Grade II listed building, though the council is unlikely to reject the request.
A brief statement attached to the planning application said simply it was starting the process of a “handover back to the landlord.”