Concerns are growing over the financial stability of the providers of the Kaupthing Edge and Icesave accounts.
Icelandic banks have attempted to reassure their UK customers that their savings deposits are not under threat.
Both Kaupthing and Landsbanki - the Scandinavian country's two largest financial firms - operate popular high-interest accounts through their British businesses, Kaupthing Edge and Icesave. However, the recent worsening of the credit crunch has hit Iceland's economy hard.
The government has been forced to rescue the third largest bank, Glitnir, from collapse. Iceland's economy is also very dependent on its financial services sector - and has therefore been more affected by the recent general downturn in investor confidence which has knocked banks' share prices and capital-raising abilities.
Indeed, both Kaupthing and Landsbanki are many times the size of Iceland's economy - leading to analysts' fears that the government cannot save them from going out of business through nationalisation.
Further complicating the situation is the fact that the two banks are under different deposit guarantee schemes, which insure customers against their bank's collapse. Kaupthing Edge is covered by the UK's guarantees, which covers the first £50,000 in a savings account from tomorrow - while Icesave customers would have to claim their first £15,000 from the Icelandic compensation scheme and the rest from the UK if Landsbanki collapsed.
Speaking to the BBC Mark Sismey-Durrant, chief executive of Icesave in the UK, said: "We have a strong capital base and 63 percent of our balance sheet is funded by deposits. We maintain strong liquidity levels."
He added that savers' money remained safe with the bank.
Tryggvi Herbertsson, economic adviser to the Icelandic prime minister, also aimed to reassure nervous savers. "The banking system in Iceland is very large compared to the economy, but still we think we can maintain the problem because the balance sheet of the bank is very good," he said.
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