Kent Reliance Building Society says it would not be treating its savings customers fairly by cutting its standard variable rate (SVR).
Many building societies are failing to put their members first by passing on the Bank of England's recent base rate cut.
Nationwide will cut its base mortgage rate by 0.3% from 1 November, well below the 0.5% cut made by the Bank of England.
The Bank of England may set interest rates as low as 2% by the end of 2009 to prevent a dangerous deflationary cycle, according to Tim Drayson, economist at Legal & General Investment Management (LGIM).
The number of mortgage products available in the UK has fallen to its lowest level since the onset of the credit crunch, with total numbers down 12.5% since August, according to Moneyfacts.co.uk.
Stonehaven has cut interest rates for the second time in as many months on its equity release products for the under 70s.
Consumer confidence has fallen further in September after remaining stable through the summer, according to Nationwide's consumer confidence index.
One in ten borrowers in the UK have missed a repayment on their personal loans, mortgage or credit cards over the past six months.
A £40-50bn taxpayer-funded life-belt for high street banks is being considered by ministers as a last resort, according to a report in The Times.
The European Central Bank (ECB) has kept interest rates stable at 4.25% despite growing pressure for a rate cut.