Northern Rock has withdrawn a number of savings deals to new customers to ensure it does not abuse any competitive advantage.
The nationalised bank is one of few financial institutions that has a 100% guarantee for the safety of customers' savings, as it is government-backed.
On Thursday, it said that it had seen a "sizeable inflow" of deposits in recent days owing to financial turbulence.
It must cap its market share of UK retail deposit balances at 1.5%.
This commitment was made as part of the nationalisation deal.
The move comes as the Irish government confirmed legislation to guarantee the safety of 100% of savings at six institutions.
Savings shift
There have been widespread reports of savers moving their savings to financial institutions regarded as secure.
But savers setting up new accounts will generally have to wait a few days before they can put money in.
"There has been a flight by some British savers to the safety of Irish banks"
National Savings & Investments - the government's savings scheme - has a 100% guarantee, in the same way as Northern Rock.
A spokeswoman for NS&I said its call centre had seen "an increase in volumes of calls" in the last few weeks, but did not provide figures to show the level of the increase.
The Anglo Irish Bank said it had received a lot of calls from savers following the Irish government guarantee.
And a spokeswoman for the Post Office, which has savings accounts run by the Bank of Ireland, said it was "continuing to see a increase in savings customers".
In the UK, only the first £35,000 is guaranteed but Prime Minister Gordon Brown said this was planned to be raised to the first £50,000.
Official figures for the flow of money out of accounts on Wednesday are unlikely to be available for another 10 days.
UK payments association Apacs said that no significant peaks had been spotted on transactions through the Chaps transmission service.
This service allows money to be transferred from one account to another in a day and is generally used for large sums of money.
The system that underpins internet and telephone banking did not show any big spikes either, Apacs said.
Rock cuts
Northern Rock cut its Silver Savings, Silver Savings 30, Business Reserve and a range of fixed rate bonds on Thursday.
This follows the recent withdrawal of its Fixed Rate Asset Bond and online E-Saver products.
"I know it is selfish but I am concerned about what is happening"
"The self-imposed framework is designed to ensure Northern Rock does not take unfair advantage of government support during the period of temporary public ownership," a spokesman said.
When it was nationalised, Northern Rock said it would limit its share of retail deposit balances to 1.5% of the market in the UK and 0.8% in Ireland. Its share of gross new mortgage loans would also be no more than 2.5% of the UK market, it said.
A number of media commentators advised savers to turn to Northern Rock at the weekend. Northern Rock does not advertise its 100% guarantee.
Moving savings
One customer who has decided to move around his savings is retired accountant David Roots, from Somerset.
The 43-year-old had £102,000 saved with ING, £52,000 with Bradford & Bingley and £21,000 in an Isa with Alliance and Leicester.
He has decided to keep £35,000 with B&B and A&L owner Santander, another protected £35,000 with ING and the remaining £105,000 with the fully protected Anglo Irish Bank.
"I know it is selfish but I am concerned about what is happening," he said.
"If people move money around then it might destabilise the banking sector, but you have to act for yourself."
He called for the UK government to protect 100% of savings, and for more regulation to prevent banks lending beyond their means.
He is one of relatively few people who have more than £35,000 saved in one account.
Figures from the British Bankers' Association published last year suggest that 4% of the 150 million deposit accounts in the UK had funds in excess of the current guaranteed safety threshold of £35,000.
Some 2% of these contained funds in excess of £50,000.
However, these figures pre-date much of the banking crisis and it is highly likely that many of these savers have since spread these funds among various accounts.
Small businesses are also covered under the UK scheme for the first £35,000 of deposits.
They must satisfy two of the following three criteria: a turnover of not more than £6.5m, a balance sheet total of not more than £3.26m and a total number of employees of not more than 50.
Larger businesses are generally excluded from the scheme.
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