A Panelbase survey of 1000 adults put support for independence on 37% while backing for the Union was 47% but those undecided were on 16%.
Among those certain to vote, support stood at 44% for independence and 56% against. But when the undecideds were asked if they had to vote today how would they choose, the gap shrank to just six points, putting Yes at 47% and No at 53%.
A separate ICM poll gave the No camp a stronger lead, putting support for independence at 32% and for the Union at 49% with don't-knows as high as 19%.
Among those certain to vote, support for independence was, therefore, 40% compared to 60% backing the Union. Yet the lead again narrowed, this time to 10 points, when people were asked to assume a No vote would not guarantee extra devolved powers for Holyrood.
Responding, Ms Sturgeon, the Deputy First Minister, insisted the Yes camp was full of optimism, noting how "one poll (was) showing that people who are now undecided are moving towards the Yes campaign and, secondly, if people see a No vote as being a vote for no change, which it most certainly is, then equally they are more likely to switch to being Yes voters. "So," she added, "with a year to go there is all to play for and there's a great deal of optimism on the Yes side."
Blair Jenkins, chief executive of Yes Scotland, claimed the poll results were evidence the momentum was with the independence movement.
"When people consider the consequences of a No vote next year, support for Yes increases. We already know people are more likely to vote Yes if they think there will be another Tory government or the UK will leave the EU," he said.
But a spokesman for the Better Together campaign insisted every single poll taken had shown a clear lead for the anti-independence argument: "The Nats look at the polls and ignore the fact that the majority of Scots remain deeply sceptical about their proposals.
"We are determined to reach out to not only those who support us, but also to those not committed. Unlike the SNP, we will not take people for granted."
Meanwhile, the row over Alex Salmond's call to have a head-to-head TV debate with David Cameron on St Andrew's Day rumbled on with Scottish Labour picking up on comments made by the First Minister.
Drew Smith, Scottish Labour MSP, said: "Alex Salmond would rather debate [with] someone who doesn't have a vote in this referendum than someone who has. Indeed, he refers to David Cameron, who doesn't have a vote as the 'organ grinder', presumably suggesting that Alastair Darling and any other Scot who has a vote is a monkey.
"This is deeply offensive for the First Minister of Scotland to refer to Scots who have votes in this referendum as monkeys," said Mr Smith.
"Mr Salmond's negativity has now descended into base insults and he needs to understand Alex Salmond does not speak for Scotland on this issue. Mr Salmond wants people to think this referendum is a battle between Scotland and England. It is not, it is Scotland versus Mr Salmond and Scotland is going to win."
In a separate development, Ms Sturgeon confirmed the Scottish Government's eagerly awaited White Paper on independence would be published in November, saying it would be a "comprehensive guide".