The plans have been lodged by Alan Soare, of Whitby, for the old Skipton branch located at 5 Coldyhill Lane, Newby.
The branch closed in 2017 after the society said it had seen a significant decline in the number of people using the branch as customers make greater use digital and telephone services.
Now the building could become a dental practice.
The plans, lodged with the council, state that the application is for “the conversion of office space into a two surgery dental practice, with decontamination room, reception, waiting area, disabled access toilet and staff area with internal work only needed to transform into an eight roomed premises for this use.”
The plans are now out to consultation.