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The watchdog announced that the government's Revenue & Customs department (HMRC) is looking at the accounts of Britain's 6,500 "high net worth individuals" to identify any tax which has not been voluntarily declared.
"High net worth individuals" are those worth over £10 million. The amounts of money at stake are huge. HMRC collected £416 million in undeclared tax from those individuals in 2014/15, and in total they contributed £4.3 billion in tax in the same period.
The statement says that £1.1 billion of the £1.9 billion HMRC is trying to recoup comes from tax avoidance schemes, which the NAO says 15% of the wealthiest use.
Numerous celebrities and prominent British figures have been caught using tax avoidance schemes in recent years. England football captain Wayne Rooney is facing a potential bill of £5 million for investing in a film project suspected to be a tax avoidance scheme, and up to 100 BBC staff are also facing bills for their elaborate financial arrangements.
In an indication of the scale of the task that HMRC faces, the report said that 6,000 enquiries it is pursuing have been open for more than 18 months, and 4,000 have been open for more than three years. That means a third of the UK's wealthiest are currently under formal investigation, with an average of four issues being examined per taxpayer.
The House of Commons' Public Accounts Committee (PAC) said the news underlined the government's lax approach to collecting tax from Britain's wealthiest individuals.
Labour MP Meg Hillier, the chair of PAC, said: "This low prosecution rate underlines concerns that high net worth individuals and their advisers are able to keep one step ahead of HMRC.
"We will be questioning HMRC about the level of prosecutions and whether it is taking the right approach."
Amyas Morse, head of the National Audit Office, said: "The tax affairs of the wealthiest in society are complex, making it harder for HMRC to ensure that they are paying the right amount of tax.
"HMRC's specialist team gives it a better understanding of the tax affairs and behaviours of these taxpayers. While the yields from HMRC's work in this area have increased it needs to evaluate what approaches are the most effective and to understand the outcomes it achieves," she added.