Commissioned by tycoon Sir Tom Hunter for the launch of a project giving voters impartial information about their options, the survey suggests a five-point swing toward independence among decided voters since the autumn.
The TNS poll put support for independence at 29%, with support for the Union at 42%, and 29% of voters saying they were undecided. However, leaving aside the "don't knows", support for Yes among decided voters was 41% - up from the 36% recorded by TNS in October.
The poll also found 69% of voters believed David Cameron should debate the key issues with First Minister Alex Salmond, something the Prime Minister steadfastly refuses to do.
Although support for a debate was highest among SNP supporters - who hope Cameron's appearance would backfire and boost the Yes vote - 60% of those voting No also wanted to see a debate, including a majority of Tory voters. Only 17% of those polled were against a debate.
TNS found 56% felt they lacked the information to make up their mind, and that only one in seven had read any of the White Paper on independence: half of those aged 16 to 24 had never even heard of the White Paper, published in November.
Hunter said people's hunger for information was behind the launch of ScotlandSeptember18.com, and said it would try to inject "momentum, data and evidence" into a debate "too important to leave to the politicians".
Stressing he would not use his fortune to influence the vote, nor declare his voting intention, he said: "It is my belief the referendum is way closer to call than anyone thinks and for that reason I believe many Scots need far more data in order to make an informed choice.
"This poll vindicates that position, with 56% of adults in Scotland saying they do not have enough knowledge about the issues related to independence to decide how to vote. Undoubtedly, this is the biggest decision we will take in 300 years and to do so Scots need answers, not rhetoric.
"Personally, I concluded I didn't have enough information to make up my own mind, hence I've established this entity to attempt to throw as much evidence-backed light on the subject as possible."
Hunter said he hoped to secure papers from Nobel laureates and other "unimpeachable" sources.
Tom Costley, head of TNS in Scotland, said: "The results show that many feel they are lacking information, particularly those currently undecided about how they will vote. There is a clear need for more and better information from both sides on those issues of most relevance to people."
Last week, an ICM poll put support for Yes at 47% among decided voters. Although the TNS poll was conducted just before the ICM survey, Professor John Curtice of Strathclyde University said it was still significant as TNS polls have, until now, made rather bad reading for the Yes camp. That TNS was now seeing movement - and had "Yes" at more than 40% for the first time among decided voters - suggested real change. Curtice said: "If you take the end of September as the baseline this is a five-point swing. There's a gradual but persistent trend." He said the shift could now help the Yes side raise more money, but could also prompt a surge in cash to the No side from worried Unionists.
Yes Scotland chief executive Blair Jenkins said momentum was firmly with the Yes camp, adding: "The most significant figure in this poll is the large number of undecided voters at 29%. This is three points below what TNS found in December and we know that the more people learn about independence the more likely they are to move to Yes."
Better Together campaign director Blair McDougall said: "We are encouraged that this poll shows that support for breaking up the UK is below 30% and support for remaining in the UK is up. However, as is the case with all polls, the only message we take from it is that we have to fight for every vote between now and September. Anyone who thinks that this referendum has been won already is mistaken."
Deputy First Minister Nicola Sturgeon said: "The gaping hole in the debate is the lack of any answers to the questions we have posed to the anti-independence campaign about what would happen to Scotland if the vote is No, in terms of more cuts to Scotland's budget, Westminster's dismantling of the welfare state, and the danger of being dragged out of Europe.
"The big finding is that people want David Cameron to debate with the First Minister Alex salmond. The Prime Minister cannot possibly get away with the hypocrisy of his position, whereby his Tory-led government is directing the No campaign from Westminster, yet he doesn't have the courage to debate Alex Salmond face-to-face."
A UK Government spokesman insisted it was providing voters with ample information, with 10 "Scotland Analysis" papers out so far.