Coutts & Co is carrying out a review and said clients would be compensated in any case where they were found to have suffered financial detriment. It pledged appropriate action would be taken.
The bank pointed out it was one of a number of wealth management providers who had been approached by the FSA back in 2011.
Coutts agreed with the FCA to conduct a review of the suitability of its investment advice given in the UK regarding investments held by clients as at the date of implementation by Coutts of the Retail Distribution Review on 26 November 2012.
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Michael Morley, the chief executive, said: "Looking back, there have been some instances where the advice given during our previous advice process could have been better, and we are working hard to address that.
"We want our clients to be absolutely certain that every investment made by them is indeed suitable, and continues to be suitable.
"If not, we will ensure that portfolios are appropriately adjusted, and if clients have suffered any financial detriment, they will be compensated in full.
"The trust of our clients in the stewardship of their wealth is our number one priority and our review is designed to make sure that suitable advice was given and put things right when it was not."
He added the bank is proud of its new Wealth Management Advice Service, which has come in following RDR.
Prior to RDR frontline staff simultaneously carried out general banking and wealth management advice. But since November 2012 created separate roles for private bankers and wealth managers.
Coutts said in a statement: "We now give objective advice to clients based on a comprehensive understanding of their personal assets and liabilities.
"Post RDR our service is a holistic needs-based service, which we operate to QCA level 6 (industry standard is level 4), supported by specialist advisers in areas such as retirement, estate and IHT planning. Our wealth managers today are all trained and tested to the highest standards."
It added its "previous weakness" lay in mainly paper-based record-keeping, but now client records are captured and held, both electronically and in writing.
The final cost of compensation is not yet known and the review is set to conclude early next year.