Union bosses are discussing a possible ballot for strike action at the mail centre in Swan Valley, Northampton.
The Communication Workers Union (CWU) has said it will meet next week to decide whether to ballot members.
Gareth Eales, local CWU branch secretary, said: “Managers are refusing to properly engage with the union over a number of longstanding issues, including planned cuts to overtime. In effect Royal Mail are not recognising the union so we need to meet to discuss how to respond.
“If members vote to strike, mail will not be delivered in Leicestershire, Coventry, Warwickshire, Milton Keynes and Northamptonshire.”
Mr Eales added: “One of the bosses told us ‘the train is leaving the station, you either get on board or you get left on the platform.’
“They’re saying do what we say, you have no choice.”
Royal Mail said it had not been notified of any possible industrial action.