The poll by Opinium, conducted last Friday after a week where the question of a second independence vote hit the headlines amid a major Labour row, found that 41 per cent of those asked agreed that the UK government should not block a referendum.
The figure was highest in London at 48 per cent, followed by 46 per cent in Scotland.
When asked if the UK government should prevent a second vote, 32 per cent overall agreed - with the highest number in Wales at 40 per cent - and 34 per cent of voters agreeing in Scotland.
However 22 per cent said they did not know on either question, and five per cent said "none of these".
The poll comes as the Labour party is mired in internal fighting over whether the UK government should block a second referendum request, after Shadow Chancellor John McDonnell said a Labour government would not do so, but Scottish Labour leader Richard Leonard re-affirming his stance against the idea.
McDonnell said: "We would not block something like that. We would let the Scottish people decide. That's democracy.” But last September Mr Leonard told the Labour party's UK conference that the next manifesto would include a commitment to oppose a second referendum on independence, and this week he said he had "made it clear" to Mr McDonnell that was still his position.
Today, writing in a Sunday newspaper First Minister Nicola Sturgeon said Mr McDonnell's statements this week on the Edinburgh Fringe had been a "refreshing change" and he had accepted "that it is the Scottish Parliament and the people who should decide on the question of whether to hold a new referendum on independence and that Westminster should not block that decision."
Scottish Conservative leader Ruth Davidson, also writing in a Sunday paper, called the internal war in Labour over the question an "utter shambles".
And former Prime Minister Gordon Brown has also warned today that "unionism appears to be sleepwalking into oblivion".
Commenting on the poll, SNP Depute Leader Keith Brown said: “People across the UK overwhelmingly believe that independence should be a matter for the Scottish Parliament and the Scottish people - not Westminster.
“UK voters believe it should be up to people of Scotland to decide how they are governed – not a Tory party which has no mandate in Scotland, putting our economy at risk to suit their own political ends.
“This poll shows that Labour and the Tories’ undemocratic stance of denying Scotland the right to choose their future is completely untenable. Faced with chaos and uncertainty in Westminster, it's no wonder that support for independence in Scotland is growing. And support for independence will continue to rise, in the face of a shambolic Brexit and an incompetent Prime Minister.
"The Tories can see that – which is what lies behind their utterly undemocratic move to block the people of Scotland having a say on their future."
Former Labour prime minister Gordon Brown today said that the UK, having previously been admired around the world for being inclusive and outward-looking, now presents a picture of division, intolerance and introversion.
In a newspaper column Mr Brown wrote that three weeks into Boris Johnson's premiership, "English nationalism is on the rise, the Conservative and Unionist party has been reincarnated as the Conservative and Brexit party (and) unionism appears to be sleepwalking into oblivion".
He said nationalism is pulling the UK apart, driving it towards "an unprecedented economic calamity precipitated by a no-deal exit from the EU".
The former PM wrote that Mr Johnson's recent visits to various parts of the UK have done nothing to dispel the impression that under him the world's most successful multinational state "is devoid of a unifying purpose powerful enough to hold it together and to keep four nationalisms - Scottish, Irish, English and also a rising Welsh nationalism - at bay".
Mr Brown also blasted the SNP, saying it is peddling what it claims is a progressive, pro-European Scottish nationalism while ignoring what he says are the hundreds of thousands of jobs at risk if Scotland leaves the UK.
He said the ideals of an inclusive outward-looking Britishness "could not survive the divisiveness and chaos of a no-deal Brexit" and to prevent the rise of dysfunctional nationalism, Mr Brown said, "the first step is to stop no deal in its tracks".