The move means people in the town could face having to drive into Exmouth and pay for parking to collect undelivered mail.
Royal Mail has said it is looking into the possibility of a ‘local solution’ for collecting mail, which could include a pick-up point.
The service said there would be no job losses and it would work with staff and unions to give them the opportunity to discuss the ‘implications’ of the move. No date had been set for the closure while staff are being consulted.
A statement said: “This move is part of our ongoing transformation and modernisation of our mail processing and delivery operations across the UK.
“This move is driven by the changes in the mail we handle – with more parcels and fewer letters in our network.
“When the move goes ahead, there will be no impact on deliveries to our customers. The postmen and women who work at Budleigh Salterton delivery office will continue to serve the local community, delivering a high-quality service to our customers.”
Royal Mail said customers would have the chance to arrange for their missed mail to be re-delivered, free of charge, on a day that was convenient for them. Customers could also arrange for a parcel to be delivered to another address within the same post code.