Around 4pc of RBS' retail and small business loan book is comprised of business with Scottish consumers and firms.
In the event of a Yes vote, it remains unclear what currency an independent currency Scotland would use.
Credit Suisse research has estimated that Scottish bank assets would stand at 12 times the nation's GDP, if it were to become independent.
In 2007 Iceland's banks held assets of less than eight times the country's national output, a multiple that was considered very high.
If a newly independent Scotland were to continue using the pound, as part of an informal currency union, it seems unlikely that the state would be able to support either firm in the event of a bank run.
"RBS believes that this is the responsible and prudent thing to do and something that its customers, staff and shareholders would expect it to do".
Its shareholders include the UK government, which owns more than 80pc of the bank's equity.
“Lloyds Banking Group has seen an increased level of inquiries from our customers, colleagues and other stakeholders about our plans post the Scottish referendum on Thursday, September 18,” Lloyds said in a statement on Wednesday night.
"While the scale of potential change is currently unclear, we have contingency plans in place, which include the establishment of new legal entities in England", it said.
According to Chirantan Barua, an analyst at Bernstein, it could cost RBS and Lloyds £1bn each to move their operations.
"The fiscal, monetary, legal and regulatory landscape to which it [RBS] is subject" in the event of a Yes vote also remains unclear.
The company stressed that "a vote on independence is a matter for the Scottish people", but that as Scotland had been RBS' home since 1727, it "intends to retain a significant level of its operations and employment in Scotland", even if it does secede.
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