The YouGov poll for The Times newspaper found backing for a Yes vote in a second ballot on the country’s constitutional future stands at 44%, with No on 56%.
It comes as the SNP concludes its national survey which aimed to speak to two million voters about their views on Europe, Brexit and independence.
Valuable data gathered during the exercise could be used by the party if Nicola Sturgeon decides to call another referendum on leaving the UK.
She has said the option remains on the table should the UK’s Brexit deal fail to protect Scotland’s interests in Europe.
Ms Sturgeon is considering arrangements which could allow Scotland to remain in the European single market, and is expected to publish options in the next few weeks.
The YouGov poll also reveals that only 31% of Scots want the Scottish government to campaign for independence in the next two years. A further 56% think it should not, while the rest do not know.
Respondents were asked about whether they think it would be “realistically possible” for Scotland to remain part of the EU after the rest of the UK leaves, regardless of their support for such a move.
Only 22% said they thought this would be “probably possible”, with 62% saying “probably not possible” and 16% said they did not know.
YouGov polled 1,134 people from November 24-29.
Scottish Conservative leader Ruth Davidson said: “The SNP has spent the last 12 weeks promoting a so-called national survey to convince the people of Scotland of the need for a second referendum.
“Today, it got its answer – fewer than one in three want another referendum, the lowest since the 2014 vote.”
She said Ms Sturgeon should “stop disrespecting the majority of the people of Scotland” with proposals for another referendum.
But the SNP highlighted the increase in support for remaining in the EU shown in the poll.
The party’s business convener Derek Mackay said: “It’s clear that people in Scotland want to leave the EU even less than they did in June when the result was an overwhelming vote to Remain.
“People are rallying around the SNP’s commitment to protect Scotland’s place in Europe – they don’t want to see us taken out against our will by the Tories.”