Yorkshire and Clydesdale bank's owner, National Australia Bank, is to shut its UK final-salary pension scheme to new members.
New staff will from next January be barred from the scheme, which guarantees a set income on retirement.
Banking union Unifi is concerned that the scheme's 13,800 existing members, currently unaffected, could lose out.
The union is threatening industrial action, if existing members' pension rights are affected in the future.
Sandy Boyle, deputy secretary of Unifi, said: "The mood of staff is such that this anger will boil over if the company moves against the rights of current members. This could include industrial action."