The venerable Coutts bank, home of the Queen’s personal accounts, has been drawn into the mounting controversy about private banking with an admission that its Swiss arm is under investigation for aiding and abetting tax evasion.

Royal Bank of Scotland, which owns the 300-year-old Coutts, admitted on Thursday that German prosecutors are investigating current and former employees of the private bank’s Zurich and Geneva offices. The bank looks after £20bn belonging to 32,000 international customers including sovereigns, celebrities and multimillionaire entrepreneurs. Each must show they have at least £1m in free cash to open an account – and even then they may not be allowed in.

Details of the Coutts investigation follow the leak of details of 100,000 accounts at HSBC’s Swiss banking business. That leak, which showed how HSBC clients might evade tax by moving their wealth around in “bricks” of cash and how the bank provided advice on tax avoidance, has lifted the lid on the usually secretive world of Swiss banking. The chancellor, George Osborne, has now promised to include in next month’s budget a new law to target those who aid or abet tax evasion.

Ross McEwan, the chief executive of RBS, admitted Coutts was facing an investigation by Germany and said private banks had been too slow to clean up their businesses following the 2008 crisis. He said: “Any situation like this we take seriously … it is the reputation of our business. This is what has tarnished the banking industry and in my view private banks have taken far too long to catch up with the public’s expectations.”

The disclosure of the investigation raises the prospect of an institution bailed out by UK taxpayers in 2008 facing penalties for helping its clients deprive other countries of tax revenue.

Coutts was founded in 1692 and still operates under the logo of the three gold crowns which the Scot John Campbell used as the bank’s sign when he set up business on the Strand in London. It has been owned by RBS for 15 years.

An account carries cachet and is a symbol of wealth. Customers are promised “the provision of exceptional wealth advice”.

Coutts’ Swiss operation is already one of more than 100 banks which are being investigated by US authorities over possible tax avoidance and RBS said the German authorities had been involved more recently.

McEwan had put the international operations of Coutts up for sale before the German investigation was revealed and on Thursday the Coutts chief executive, Rory Tapner, departed suddenly. The bank said his exit was part of a wider management overhaul.

McEwan said the business had been put on the market because it didn’t make money and he told ITV there were moral as well as business reasons for doing so. “I want to be very clear if we find anything that has evidence of wrongdoing we will come down incredibly hard on any of those issues,” he said.

RBS intends to keep Coutts’ UK business, which is steeped in history. Ledgers kept in the bank’s Strand boardroom contain signatures from famous past clients, including Charles Dickens and Bram Stoker, author of Dracula.

But the bank has not escaped all controversy. After the financial crisis the founder of the Nectar loyalty card scheme, Sir Keith Mills, started a campaign against Coutts after he put his money in savings bonds issued by AIG, the troubled US insurer. Coutts denied it mis-sold the bonds. Last year the bank admitted that thousands of its well-heeled customers may have been given unsuitable investment advice and it was reviewing files going back as far as 1957.

The Geneva operations have only existed since 1987 but are now adding to the litany of conduct issues facing RBS, which is still 81% owned by the taxpayer. It has already been fined for rigging interest rate and currency markets and faces nearly £2bn of penalties for the way it sold mortgage bonds in the runup to the 2008 banking crisis.

The disclosure of the German investigation came deep in the pages of legal warnings attached to RBS’s annual financial results which showed the bank lost£3.5bn last year. It was the bank’s seventh consecutive year of losses, which now total £49bn since the financial crisis, when the government used £45bn of taxpayers’ cash to stop the Edinburgh-based bank collapsing in 2008.

Following the Guardian’s revelation that Stuart Gulliver, the boss of HSBC, has his own Swiss bank account which is routed through Panama, McEwan also faced questions about his banking arrangements. McEwan said: “I’m a proud Kiwi but I work here and I pay taxes here.” He said he had accounts in New Zealand, Australia and the UK, but did not have a Swiss account.