It said it wanted staff at both sites to work at the new centre, but union leaders said many workers would not be able to afford to travel or relocate. They warned the planned closures would have a "devastating" impact on local economies.
Up to 2,000 workers could walk out after 60% of those who voted came out in favour of industrial action. Mick Kavanagh from the union said he hoped the strike could still be avoided.
"It really is down to Royal Mail to get around the negotiating table this week and get this sorted out," he said.
Saturday is the last first-class posting day before Christmas and the union has promised that workers will do all they can to clear the backlog in time for the holiday.