Ministers and members of the royal family have been lobbying the prime minister for a royal yacht since September, even though Downing Street insisted government support was conditional on no public money being made available.

Downing Street said the prime minister was happy to facilitate discussions, as government officials released letters showing that both the higher education minister, David Willetts, and the education secretary, Michael Gove, had been lobbying the prime minister to back the plan.

A letter from Willetts to the prime minister claims the idea has the support of both the Prince of Wales and Princess Anne. The plan for the yacht is the brainchild of Rear Admiral Bawtree.

Gove angrily denied he supported any public funding, although a letter leaked to the Guardian at the weekend showed he did see public funding as the chief option.

Ministerial sources also conceded that Gove was concerned the diamond jubilee could be overshadowed by the London Olympics and he was anxious to promote celebrations for the Queen this year.

Willetts wrote to Cameron in September with details of a "future ship project for the 21st century" being drawn up by Bawtree and stressing no public money would be available.

The proposed ship would be made available for trade and business events, and be a potential replacement for the royal yacht Britannia.

"The Rear Admiral considers it could be used as a training resource for young people and could be made available to research funders as a research vessel," Willetts wrote in his letter to Cameron. He asked Cameron to write to Bawtree to say he believed the idea was worthy of endorsement.

Gove wrote to the prime minister on 12 September again supporting the project: "I believe that approving this ship to become a royal yacht would be an excellent way to mark the Queen's diamond jubilee and to thank her as a nation for her long and untiring service to this country."

In this letter he stated: "No money should be made available from the public purse", but in a second letter dated 11 December he did not make this point: "My suggestion would be a gift from the nation to the Queen thinking about, for example, David Willetts' excellent suggestion for a royal yacht – and something tangible to commemorate this momentous occasion. If there is not sufficient public money available then we could surely look for a generous private donation, for example, to give every school child a lasting memento of the occasion or possibly to allow every school to buy a permanent reminder."

Education department officials said the letter was loosely worded.

Both the royal family and Downing Street will be upset their plans for a royal yacht replacement have emerged in this way.