Prime minister throws weight behind £60m-project despite protests that taxpayers should not have to foot the bill

David Cameron has swung behind plans for a new, privately funded royal yacht that will double up as a university of the seas, and provide accommodation for royalty in the ship's stern.

Cameron has endorsed the idea after lobbying from the higher education minister David Willetts and the education secretary Michael Gove. The idea, at one point described by Gove as a gift from the nation to the Queen on her diamond jubilee, also has the backing of the Prince of Wales and Princess Anne, according to letters sent to the prime minister by Willetts.

Downing Street sources said the prime minister regarded the idea as excellent, and discussions have been held with Portsmouth city council for the yacht to be berthed in the south coast port.

The £60m yacht has so far found £10m in backing from financial leaders in Canada and an unnamed £5m private donation.

Cameron's enthusiasm comes despite a storm of protest after the Guardian revealed ministers had recently discussed taxpayers paying for the yacht as a "present" to the Queen – prompting critics to accuse the government of being out of touch with the nation's economic priorities.

After the leak of a December-dated letter from Gove lobbying fellow cabinet ministers for a royal yacht to the Guardian on Sunday, government officials have released further letters showing that ministers have been urging the prime minister to back the plan since September. The plan for the yacht is the brainchild of Rear Admiral David Bawtree, a former naval base commander in Portsmouth.

In the Commons on Monday Gove angrily denied he supported any public funding for the yacht, although the letter leaked to the Guardian showed he did at least at one time see public funding as the chief option. The education secretary's office insisted that Gove had, in an earlier letter, rejected the possibility of public funding.

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, it emerged on Monday, 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, Willetts wrote.

"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 writing instead: "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, especially since the idea of a yacht being built for a royal family sits at odds with a year of great austerity.

But advocates of the plan claim the 600ft ship can be built with private money, and its upkeep paid for by fees from university students staying on the ship. The plan is to provide education and training to about 200 young people over the age of 16 for periods of three months. In addition there would be a research element for scientists.

There will also be concern over whether government officials are being drawn into endorsing an essentially commercial idea that needs private backing. There would also probably be a public cost for the upkeep of the yacht, as well as security costs. It is not clear if fees from students could cover these costs. The proposals have the enthusiastic backing of the Liberal Democrat controlled Portsmouth council.

The final royal yacht was decommissioned in 1997 by the Labour government on grounds of cost.

• This article was corrected on 17 January 2012 to delete a line saying that Foyles had offered to contribute £500,000 for books for a library on a revived royal yacht. A Foyles representative said the company was unaware of such an offer.