NS&I has occupied the site at Cowglen since the then National Savings Bank was decentralised from its London headquarters into four units in the early 1970s, with other sites at Blackpool and Durham.
The Glasgow office was built to house 6000 people but currently there are only 650, involved in processing transactions and answering customer enquiries. Half of the building is unused.
The state-owned group had to choose between a major refurbishment or finding another premises, and confirmed it is looking for a new home in the city centre.
It is understood to have narrowed down its options to six existing buildings with a view to moving in March 2014.
NS&I insisted there would be no change to the number of Glasgow employees. According to industry sources, however, the company's requirement is for between 40,000sqft and 60,000sqft.
Although NS&I would not comment on this, the commercial property industry works on a rule of thumb of 100sqft per person. On that basis, the requirement will be seen as somewhat low.
Alan Brown, full-time officer at the Public and Commercial Services union, said it had not been consulted and was not expecting a decision until the new management contract for NS&I is awarded in April. It is currently held by Atos.
He said: "We are waiting to see the proposals. If it's an alternative site close by and it's somewhere that the staff can travel to within a reasonable time, there's not going to be any issue."
A source in the property industry said the Cowglen site was likely to attract "substantial interest" from the market if it is put up for sale, and would be particularly attractive to housebuilders.
A spokesman for NS&I said the process was at a "relatively early stage". He added: "We have decided not to go ahead with redeveloping Cowglen and are actively looking at other options."