The bank warned more than one million of its customers it could introduce negative interest rates if that were to happen.
This would mean customers are effectively paying the bank to hold their savings.
A letter said the change would only affect business and commercial customers.
It said: “Global interest rates remain at very low levels and in some markets are currently negative. Dependent on future market conditions, this could result in us charging interest on credit balances.”
A spokesman for the taxpayer funded bank, whose parent is the Royal Bank of Scotland, said there were no current plans to charge personal account holders.
But there are fears the move could see savers remove their money from the bank – and that other banks could follow suit.
“The danger is many people will just think, I’m going to put the money under the mattress. That could have security risks, especially for older people.
“You don’t want your life savings out of the bank, you want them somewhere safe.
“But if the bank is going to charge you for keeping your money, and every day you have it there it is worth less and less, you can see why people would say, I’m not going to do that.”
Earlier this month, the Bank of England’s Monetary Policy Committee voted to keep the base rate at 0.5%, where it has been since March 2009.
But Bank of England governor Mark Carney has hinted that the central bank may slash interest rates over the summer months from this already historic low.