Thousands of savers with a Cypriot bank with branches in London and Birmingham will not have the protection of the UK's compensation fund if Cyprus goes under.
British customers of the Laiki Bank are instead reliant on the Cyprus Deposit Protection Scheme, which covers up to 100,000 euro, or around £87,000.
A spokeswoman for Laiki Bank UK said there is "no sign" of its 13,000 depositors panicking about their money and insisted it is "business as usual" for the bank. She added: "We are sitting on a sea of cash. Withdrawals have been very, very low."
Those with money held at its three branches in London and one in Birmingham are not covered by the UK's Financial Services Compensation Scheme (FSCS), which protects up to £85,000 of customers' money if a financial institution goes bust.
The spokeswoman for the bank, which has branches in Mayfair, Palmers Green and Finchley in London and in Erdington in Birmingham, said that anyone who has concerns should get in touch. She added: "It is business as usual. Customers have been dealing with the bank in the normal way. There is no panic."
Pressure is mounting on Cypriot politicians to agree a plan which will convince international lenders to stump up the money the country needs to avoid bankruptcy within days.
Cyprus needs to have come up with a roadmap by Monday as the European Central Bank has said it will cut off emergency support to the banks, which could trigger their collapse and end up with the country being forced to leave the euro.
Under European law, all European Economic Area countries must provide 100,000 euro in protection for the total deposits of each eligible saver.
The Cyprus Deposit Protection Scheme has been in operation for 13 years and it would come into play if a determination was made by the Central Bank of Cyprus that the bank is unable to repay deposits. In theory, payments would then be made within 20 working days, although the Central Bank of Cyprus can extend the repayment period by up to 10 working days.
Prime Minister David Cameron discussed the financial crisis in Cyprus with Russian president Vladimir Putin in a phone call on Friday morning.