The twist will be that the offer will be time-limited – the First Minister will have to hold a poll before August 2013.
The implication is that if he fails to take it up, then Westminster will seize the initiative and hold its own referendum.
The Prime Minister's move raises the constitutional stakes and is a clear bid to put pressure on the SNP leader to get a move on.
It comes as a poll of 500 Scots, conducted by Ipsos Mori for the new think-tank BritishFuture, launched today, puts support in Scotland for independence at 29% compared to 54% who want the nation to remain part of the UK. Some 17% were don't-knows.
In his first interview of 2012, Mr Cameron – who earlier made it clear that, as an English Tory MP, he did not envisage leading the anti-independence campaign in Scotland –claimed Mr Salmond's delay in holding a referendum until 2014 or beyond was creating uncertainty. He said this was "damaging Scotland's economy because you have companies and other organisations asking, 'well, what is Scotland's future? Is it within the United Kingdom or not?'".
He also insisted: "It's very unfair on the Scottish people themselves, who don't really know when this question is going to be asked, what the question is going to be, who's responsible for asking it. We owe the Scottish people something that is fair, legal and decisive."
Once again, Mr Cameron declared his support for the United Kingdom, describing it as "one of the most successful partnerships in the history of the world" and saying it would be "desperately sad" if Scotland broke away.
Asked if he wanted the referendum to happen before 2014, the PM replied: "This is a matter for the Scottish people.
"If there are problems of uncertainty and lack of clarity, I don't think we should let this go on year after year.
"That's damaging for everyone concerned ... My view is sooner rather than later would be better."
Last night, however, the Nationalists reacted with disdain, branding Mr Cameron's move as a "power grab".
Angus MacNeil, SNP MP for the Western Isles, urged him to "butt out" and tweeted: "Westminster Eton Toff will not decide our referendum."
A spokesman for Mr Salmond said: "The Scottish Government achieved an overwhelming mandate from the people of Scotland to hold the referendum in the second half of this parliamentary term and that is exactly what we will do."
Ostensibly, the reason for the Prime Minister's move is advice from Whitehall lawyers, who have concluded Holyrood cannot hold a referendum without a substantial risk of legal challenge.
This is because, at present, only Westminster, as the constitutional authority, can hold a legally binding poll.
The fear is that any challenge in the courts could delay a referendum for months, if not years.
"While the UK Government might turn a blind eye to Holyrood holding an independence referendum, others would not," explained one Coalition source.
To remove this risk, Mr Cameron and his colleagues have decided to consider empowering Holyrood to hold such a legally-binding poll. They can do this in two ways: amend the Scotland Bill, which is due to begin its House of Lords stage later this month; or introduce a Section 30 order to the 1998 Scotland Act to extend Holyrood's powers.
The Herald has been told that, while no firm decision has yet been made on which route to take, ministers are minded to opt for the latter as it will be quicker to implement.
However, politically, the consent of Holyrood will be desirable on either route. If it is not forthcoming, a UK Government source made it clear: "The option of Westminster holding a referendum is not off the table."
Last night, Margaret Curran, the Shadow Scottish Secretary, told The Herald: "The only thing standing between the Scottish people and a referendum is Alex Salmond.
"The subject of Scotland's future is of such fundamental importance that we should have a referendum as quickly as possible. "We can't hang on forever," she added.
l An amendment to the Scotland Bill by Baroness Taylor, the former Labour Chief Whip, will seek to enable Scots living in other parts of the UK to vote in the independence referendum.
The chance of the Bill being successfully passed is thought to be nil.