The improved rates for savers in "closed" accounts - those no longer open to new customers - will be announced shortly.
The rate improvements at the Portman, the largest regional society in the South of England, will affect 80,000 of the society's 650,000 accounts.
In a similar move, 200,000 savers in closed accounts with the Bradford & Bingley were last week switched to new accounts paying higher rates, at a cost of pounds 5m a year.
The Portman's move will be presented as a demonstration of its commitment to offering members a better deal while staying a mutual. It has been critical of societies that become banks and pay windfalls, and could be one of the biggest societies left after the current rash of demutualisations.
Societies have tended to let rates on closed accounts slip below those on "live" accounts into which they are still trying to attract money. But following complaints from savers and criticism from the Building Society Ombudsman in recent years, many have already made them more competitive, or got rid of them altogether.
John Gully, head of corporate affairs at the Portman, said: "In some ways, we are worse than many." The society still has more than 100 accounts running, most of which are closed. Rob Thomas, building society analyst at City investment bank UBS, said: "You can't sing about how much you love members when you're operating these sorts of accounts."
Mutuals, goes the argument, have a special responsibility to give existing savers (and borrowers) a good deal - they own the society, after all - rather than just favouring new customers.
Mr Gully would not give details of the Portman's exact rate changes, and it is not clear how low rates are on the society's closed accounts at present. But he said the average balance in these old accounts is less than pounds 1,000 and they total less than 5 per cent of the society's savings balances.
There are no plans to follow the rate improvement with a special loyalty bonus, as other societies in the mutual camp have.
Moreover, improving closed accounts does not solve savers' wider problems. Rates are at an all-time low and continue to be shaved following the Chancellor's 0.25 per cent cut in base rates last month. This week, for example, the Portman cut the rate on its popular and competitive Instant Access account by 0.3 per cent to 4.5 per cent, while National Savings cut rates on five of its products. Meanwhile societies that have announced windfalls have been able to pare away rates knowing that savers feel locked in, fearful that they will lose out if they run down their balance.Reuse content