The sale could fetch about $1 billion for RBS, which was bailed out by the British government during the 2008 global financial crisis. Coutts International, which counts Queen Elizabeth as a customer, manages about $36 billion of which about a third comes from Asia.
RBS and its adviser Goldman Sachs have sent out sale documents containing financial and other details of the business to companies including Singapore lenders DBS Group Holdings Ltd and United Overseas Bank Ltd, Societe Generale, HSBC, Bank J Safra Sarasin, BNP Paribas, Canada's Bank of Montreal among others, the people who had knowledge of the matter told Reuters.
First-round bids are due before Christmas, they added.
Officials from Credit Suisse, Julius Baer, Maybank, BNP, HSBC, United Overseas Bank, DBS and Sarasin declined to comment.
Bank of Montreal, Societe Generale and RBS did not respond to a Reuters query. Goldman Sachs declined to comment.
Sources declined to be identified as the sale process is confidential.
To increase the competitive tension and secure a better price, RBS could break up the business into Asia and Europe, Reuters previously reported.
But if RBS decides to sell the business in one chunk, Swiss private banks are seen as frontrunners for the business, the people said.
The sale of Coutts International will help RBS, which is owned by the British government, to focus on domestic lending.
The potential sale comes after French bank Societe Generale sold its Asia private bank for about $250 million earlier this year to Singapore's DBS.