Union leaders ordered a strike ballot yesterday after a leaked document was said to recommend contracting out the majority of work carried out by the centre's 3,400 staff.
The first walk-out could be on Friday if there is a 'yes' vote.
The vote is the latest sign of discontent in the civil service over moves towards privatisation and the threat of a public sector pay freeze next year.
Bob Hart, of the National Union of Civil and Public Servants, said there was deep concern about the potential loss of jobs. 'We see this as a very serious attack on the public sector which has to be defended with a strong campaign in the same way as the threat to miners' jobs,' he said.
Stephen Curtis, chief executive of the DVLC, which is an executive Government agency, said no decisions had been taken over the future of the Swansea centre.
Civil service unions believe that thousands of jobs could disappear through the Government's 'market testing' exercises whereby private companies are invited to bid for work now being carried out in the public sector.
They are also expected to resist any attempts to restrict pay rises to less than the rate of inflation.