The government would "react favourably" to the Future Ship Project which aims to build a royal yacht for hosting royal events.
The BBC understands David Cameron has given his "full support" to the plan.
Education Secretary Michael Gove had expressed his support for a new yacht as a potential gift to the Queen from the nation to mark her Diamond Jubilee.
But he was criticised for suggesting public money could be spent on it.
The Future Ship Project for the 21st Century (FSP21) involves building a ship 600ft long, to be used for trade and business events, as well as a training ship for 200 young people.
BBC deputy political editor James Landale said Rear Admiral David Bawtree, who is behind the plan, said Mr Cameron wrote to him in October saying it was a inspirational initiative that had his full support.
Ministers say it is backed by the Prince of Wales and Princess Anne.
Mr Gove made the proposal in letters to Culture Secretary Jeremy Hunt, Prime Minister David Cameron and Deputy Prime Minister Nick Clegg last year.
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Downing Street spokesman
If there is something we can do as government to support this, we will look at this”
Fellow minister David Willetts also wrote to Mr Cameron.
Downing Street emphasised that taxpayers' money would not be used on any new royal yacht.
"Clearly there's a difficult economic situation," a prime minister's spokesman said.
"There are scarce public resources. Therefore we do not think it would be appropriate to use public money at this time."
But speaking about FSP21 later, No 10 added: "This issue is, what can we do as government to support this? If there is something we can do as government to support this, we will look at this.
"We would react favourably to that."
Mr Gove said the Queen's "highly significant contribution" to Britain and the Commonwealth should be recognised with a "lasting legacy".
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Tom WatsonLabour Party deputy chairman
When school budgets are being slashed, parents will be wondering how Gove came even to suggest this idea”
He wrote to Mr Hunt: "In spite, and perhaps because of the austere times, the celebration should go beyond those of previous jubilees and mark the greater achievement that the diamond anniversary represents.
"Events such as proms and the party at the palace organised for the Diamond Jubilee, and street parties, although excellent, are transient.
"It would be appropriate to do something that will mark the significance of this occasion with fitting ceremony.
"My suggestion would be a gift from the nation to Her Majesty; thinking about David Willetts's excellent suggestion of a royal yacht, and something tangible to commemorate this momentous occasion."
After criticism in the Commons over the expense of building a yacht - estimated to be £60m - Mr Gove aimed to clarify his suggestion.
He said that his letter had been quite clear that the Future Ship Project was one option "where no public funding should be provided".
The last royal yacht, Britannia, served the Queen for 44 years, carrying her and the Royal Family on 968 official voyages before being taken out of service in 1997.
The ship, currently in dry dock undergoing repairs, is now a tourist attraction in Edinburgh.
"The Diamond Jubilee must not be overshadowed by the Olympic Games, but form an integral part of this great year for our country," Mr Gove also wrote.
Labour Party deputy chairman Tom Watson said: "We're all looking forward to the Diamond Jubilee. The significance of the occasion should be celebrated across the country.
"But Michael Gove has shown he is out of touch with this proposal. When school budgets are being slashed, parents will be wondering how Gove came even to suggest this idea.
"This is not the time to spend £60m on a yacht."
A Department for Culture, Media and Sport spokesman said the palace had made it clear that the Diamond Jubilee celebrations should reflect the current economic climate.