An SNP-commissioned survey by Panelbase found 63% support in Scotland for a head-to-head clash, and 56% backing in the rest of the UK.
Awkwardly for Cameron, most Conservative voters outside Scotland also want a televised showdown.
Despite repeated goading, the Prime Minister has refused to debate Salmond, deferring to Alistair Darling, chairman of the Better Together campaign. However, Salmond refuses to debate Darling, producing a stalemate.
In the opening campaign salvo of 2014, Salmond has now written to Cameron saying it is "simply unacceptable" for him to duck out of a debate.
In his letter to the Prime Minister, Salmond claimed the Coalition was "itching to cut Scotland's funding in the event of a No vote" and picked up on Cameron's New Year appeal for Scotland to remain part of the United Kingdom.
The First Minister wrote: "I note you used the centrepiece of your New Year address to attack Scottish independence. It seems you want to dictate the terms of the debate about Scotland's future without taking the democratic responsibility to defend your views in open debate. That is simply unacceptable.
"It is only right and proper that myself, as head of the Scottish Government, and you as the head of the Westminster government should debate key issues as the heart of the referendum - such as Scotland's funding - so that the people of Scotland understand exactly the consequences of both a Yes and No vote. I urge you, once again, to agree to a televised debate."
The release of the letter coincided with the SNP issuing its poll findings on the subject of a televised debate.
Panelbase asked 1012 people in Scotland and 1011 in the rest of the UK last month if they agreed there should be a debate between the two leaders.
In Scotland, 63% of respondents said Yes, 25% said No and 11% were undecided. Support for a debate was highest among SNP voters at 79%, but 57% of Labour and 50% of LibDem voters also supported the idea. Only among Scots Conservative voters were a majority against a debate, 54% opposed and 35% for.
In the rest of the UK, 56% said Yes to a debate, 24% said No, and 19% were undecided. Two-thirds of both Labour and Liberal Democrat voters were in favour, as were 57% of Conservative supporters.
Accusing Cameron of being "scared" of Salmond's debating prowess, Deputy First Minister Nicola Sturgeon said: "It is not possible, with any degree of consistency or credibility, for the Prime Minister to involve himself and his Government in the referendum process and then refuse to publicly debate these very issues."
Returning fire, Better Together has released its own poll findings, showing two-thirds of Scots want better childcare from the SNP using Holyrood's existing powers, rather than it being made conditional on a Yes vote in September. Even among SNP voters, 45% backed improvements now, against 41% who said a Yes vote was needed.
Free nursery provision for three and four-year-olds will rise in the autumn from 475 hours to 600 hours a year, but the SNP says independence would allow a "transformational change" to 1140 hours for all children over the age of one.
However, a Better Together YouGov poll found that 64% of Scots wanted improvements under devolution.
Scottish Labour leader Johann Lamont said voters were "fed up" with the First Minister putting independence ahead of their interests.
She said: "The Scottish Government has the power to help parents who are struggling to juggle family life with their work commitments. Alex Salmond could act today to provide them with better childcare, but would rather use it as a false argument for breaking up the UK now."
LibDem leader Willie Rennie said: "Parents need to see SNP ministers spending less time talking about the powers they want and more time using the ones they already have to help children."
Tory spokeswoman Liz Smith said the failure to act on childcare illustrated "why the people of Scotland feel so badly let down and why they do not trust a word the SNP says".
But SNP Children's Minister Aileen Campbell said: "The increase in childcare we propose can only be delivered when the Scottish Parliament has responsibility for all spending and revenue in Scotland. If we can raise female [employment] rates … that can generate £700 million in extra tax and national insurance that helps pay for the policy."
Reacting to Salmond's letter, a Downing Street spokeswoman said: "The Prime Minister is Prime Minister of the whole United Kingdom and of course has a role in the referendum campaign.
"However, he believes the debate should be led by Scots in Scotland and that is why Alistair Darling will lead for the No campaign."