The bailed-out banking group has told staff that the international unit of Coutts is under review and options on the table are thought to include a sale, a merger or a spin-off as a joint venture.
RBS said earlier this year that it would shift Coutts into its commercial and private banking unit as part of a broader plan to streamline the business into a UK-focused retail bank.
Coutts, founded on the Strand in London by a Scottish goldsmith in 1692, has been part of RBS since the group's takeover of NatWest in 2000.
RBS Group, now 81 per cent-owned by the UK taxpayer, has scaled back the global reach of the private bank since the financial crisis.
In 2012, it sold off Coutts' Latin American, Caribbean and African private banking operations to the Royal Bank of Canada.
Chief executive Ross McEwan, who has led the group since October, is targeting a 15 per cent return on equity in all parts of the business - a goal that was met by the firm's private banking business in the first half of the year, results released on August 1 showed.
While RBS's review of its wealth management operations is ongoing and will not conclude for several months, Reuters has reported that several potential buyers from around the world have already expressed an interest in the international unit of Coutts.
Coutts & Co Limited, which contains the firm's international private banking operations, posted operating profits of 103 million Swiss francs (£67m) in 2013.
However, it was pushed to a net loss of 45.1m francs by tax settlements in the UK and United States as well as the closure of its office in Berne and credit impairments in Asia.
The company is among 100 Swiss banks in talks with American authorities as part of a crackdown by the US on its citizens avoiding tax.
Coutts' international business has around 1,250 employees and accounts for 41 per cent of the private bank's customer assets.
RBS intends to keep the British branches of Coutts. The group also sees Adam & Company, the Scottish private bank with around 7,500 clients, as a core part of its business, according to a person familiar with the firm.
Mr McEwan is also overseeing the sale of Citizens Financial, RBS's American retail bank. The company filed in May for a share offering in New York but is also pursuing a trade sale.
Shares in RBS closed down 6p at 339.6p yesterday.