The Financial Services Authority (FSA) is to change the rules in the wake of the near-collapse of lender Northern Rock, which fell victim to the credit crunch last summer.
Long queues of depositors formed outside branches of the ailing firm, now nationalised, in the first bank run in the UK for over a century.
Speaking to Bloomberg, London-based lawyer Ian Mason commented that the rule might "avoid great public concern or any run on the bank if the matter became public at that time".
He added: "This gives the banks a little breathing space whilst additional financial support is being provided by the central bank."
Meanwhile, the FSA is moving ahead in its efforts to tackle insider trading, by launching its second-ever criminal prosecution for the offence since the body was empowered to do so in 2004.
The move was confirmed by an FSA spokesperson, speaking to the Times newspaper.