The yes campaign received a boost, putting it neck and neck with the no campaign, when those surveyed were asked to consider the prospect of leaving Europe when deciding how to vote in the 2014 referendum.
The Panelbase poll of 1,004 people for the Sunday Times and Real Radio Scotland recorded support for independence at 36% when it asked "should Scotland be an independent country?".
Meanwhile, 44% answered no, and a further 20% said they did not know.
But when asked "how would you be likely to vote in next year's Scottish independence referendum if the UK was looking likely to vote to withdraw from the EU?", 44% said they would be likely to vote yes, matched by 44% likely to vote no, while 12% said they did not know.
Commenting on the poll findings, Ivor Knox, of Panelbase, said: "The prospect of the UK leaving the EU has little impact on those Scots who have formed a view on independence, but among undecided voters three times as many tend to support independence as oppose it, under those circumstances."
The poll, which was conducted between May 10 and 16, comes after the Conservatives published a draft bill, paving the way for a referendum on Britain's EU membership in the next parliament.
David Cameron wants Britain to remain in the EU under renegotiated terms which he wants to put to the public in an in-out referendum to be held by the end of 2017.
SNP MSP Clare Adamson welcomed the Panelbase poll.
"Plans for the UK to exit Europe change the terms of the debate - putting yes and no neck and neck at 44% - because the Westminster system getting dragged along by a Ukip agenda underlines that it is out of touch with Scotland," she said.
"It shows that only a yes vote puts the people of Scotland in charge of our own future."
The Panelbase survey also placed the SNP ahead in terms of Holyrood elections.
Asked how they would vote in a Scottish election tomorrow, 40% of those polled said they would give their constituency vote to the SNP, compared with 27% for Labour, 12% for the Tories and 5% for the Lib Dems.
The SNP also led the regional vote with 39%, while Labour recorded 23%, the Tories 11% and the Greens and Lib Dems both on 5%.
The Panelbase survey also comes as UK-wide support for the UK Independence Party (Ukip) has increased, according to two separate polls.
A ComRes poll for the Sunday Mirror and Independent on Sunday recorded support for Ukip at 19%, with 35% for Labour, 29% for the Conservatives and 8% for the Liberal Democrats.
On EU membership, the poll recorded that 46% of Britons would vote to leave in a referendum now, compared with 24% who would stay in. But if some powers were returned to the UK from Brussels 43% would vote to remain in the EU, with just 24% determined to see Britain leave.
Another poll by Opinium, published today, puts support for Ukip at 20%.
The party's leader, Nigel Farage, was forced to leave a press conference in a police van when he was mobbed by rowdy protesters in Edinburgh on Thursday. Mr Farage said the protest was by Scottish nationalists who shouted anti-English taunts.
His party, which is standing a candidate in the forthcoming Holyrood by-election for Aberdeen Donside, has so far failed to make in-roads with the Scottish electorate, with no elected representatives north of the border.