According to media reports today, a number of potential buyers are believed to be interested in buying the operation in full or part. Coutts was established in the 18th century and is often referred to as the Queen's bank as it is believed to handle her account.
If the deal goes through it may mean a split for Coutts into two operations, one based in the UK and one based in Zurich. The latter would run private banking and wealth management services that manage about £20bn in territories such as Hong Kong and the Middle East.
Coutts International has clients in seven countries and 1,200 staff. It could be an attractive proposition for an international private bank or wealth manager looking to expand.

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According to reports today, an internal memo has been circulated to staff which said: "We will... work with local management teams to explore options including merging the remainder of the current Coutts International business, considering joint ventures or a sale, thereby reducing RBS's footprint internationally."
It added: "There are no immediate changes for individuals in these businesses and it is important that we continue to work together to deliver for our customers."
Coutts's international operations were slimmed down two years ago after the Royal Bank of Canada bought its Latin American, Caribbean and African private banking arms.
RBS is over 80% owned by the British taxpayer following a £45bn rescue deal in 2008 saw the bank move into state ownership as the banking crisis engulfed much of the financial world.
Recently RBS has moved back into profit and it is believed it will be returned fully to the private sector in due course.