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The prestigous bank, whose customers include the Queen and wealthy celebrities such as Sir Elton John, has been accused in civil court cases in America, says the Observer.
Bank chiefs have denied the allegations made against it and deny they are being investigated.
The report says the investigation was launched after a customer at the bank's New York branch told a court that Coutts had arranged for him to attend a seminar in Monaco where he was taught tax evasion techniques. The tips would have also enabled him to launder money, the man said.
The Federal Reserve Bank and the Manhattan District Attorney's Office are both looking into the claims, the Observer says.
But Coutts says the customer only made the claims after defaulting on a loan of more than $45m and being sued for it by the bank.
According to the Observer, other serious allegations have also been made against the New York branch of Coutts in ongoing civil cases, which are not yet public.
They are said to involve "a pattern of fraudulent and improper practices as part of a misguided programme of engineering transactions to maximise its own pecuniary benefit".
'No bank probe'
Glyn Jones, chief executive of the Coutts Group, told BBC News Online the bank would never organise any instruction in tax evasion or money laundering.
"I haven't ready the civil proceedings recently but I can categorically deny Coutts would ever arrange a course at which money laundering techniques wer explained or encouraged," he said.
Mr Jones confirmed that a client in New York had defaulted on a major debt but said he did not know what the client had said about Coutts.
He also insisted the Federal Reserve Bank was not investigating.
"There's no substance whatever to the claim Coutts is being investigated," he said.
"We are seeking to recover a debt and we're being successful in the courts on it."
The New York branch opened in Manhattan in 1994 and closed in 1998, with losses of more than $50m.
Coutts, a private subsidiary of the NatWest, also counts the Queen Mother and Jonathan Aitken among its customers.
It prefers clients to have salaries of at least £150,000 or liquid assets of £100,000.
It also likes customers to keep a minimum £3,000 balance in their current accounts, or it levies extra charges.
Customers have their own personal bankers to look after every aspect of financial management, including offshore trusts and tax.
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