The paper cites a source involved with the project who claims the warning came from a former private secretary to the prince in a meeting at St James's Palace shortly after he had taken over the presidency of the charity from the Queen Mother, who died in March 2002.
It is understood that soon after he took over, he requested the power of approval over new architectural projects and asked for his advisers to scrutinise plans for the Swindon headquarters.
Clarence House would not comment on the meeting but a representative said: "What the prince wouldn't do is open or endorse a building that didn't adhere to his own principles on sustainability and design."
The threat is understood to have occurred early in the design stages after Prince Charles and his advisers had seen plans showing a triangular, low-profile structure on the site of a former Victorian engineering foundry.
As tensions mounted over the design, a meeting was held at St James's Palace between National Trust director general Fiona Reynolds and Prince Charles' private secretary Elizabeth Buchanan.
An apparent compromise was reached as plans went ahead with the addition of more than 1,000 photovoltaic panels on the roof.
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