The Progress Scotland survey found 37 per cent of voters “completely support Scotland staying part of the UK”, compared with 26 per cent who “completely support Scotland becoming independent”.
The Survation poll was commissioned by Progress Scotland, headed by former SNP Westminster leader Angus Robertson, and surveyed 2,000 Scots at the start of the month.
It sought answers on a spectrum of 0-10 to identify strength of opinion on the issue, with 0 marking “complete” support for staying in the UK, while 10 “complete” support for independence.
A narrow majority (51 per cent) were more supportive of the Union, with 6 per cent in the middle and 40 per cent more supportive of independence.
The remaining 3 per cent were classed as “don’t knows”.
The poll also found 16 per cent of voters who backed independence in the 2014 referendum now “completely” support Scotland staying part of the UK compared with just 4 per cent of 2014 No voters who “completely” support independence.
Scottish Conservative constitution spokesman Adam Tomkins said: “Far from gaining momentum, it seems the Yes campaign is losing supporters at a higher rate than it’s gaining them.”
Pamela Nash, chief executive of Scotland in Union, said: “This poll for an anti-UK organisation has backfired spectacularly, instead revealing the huge strength of support for remaining in the UK.
“No matter how Progress Scotland tries to spin the findings, its own figures show that massive numbers of former Yes voters have changed their minds and would now vote to stay in the UK.”
First Minister Nicola Sturgeon is seeking to hold a referendum on independence next year and most recent polls indicate that support for independence is neck and neck with support for the Union among Scots.
Mr Brown said: “People in Scotland overwhelmingly oppose leaving the EU, and it’s abundantly clear that views are changing on independence as voters look to escape this Brexit nightmare.
“With Westminster in chaos, and Boris Johnson in Downing Street, more and more people are ready to listen to the case for a progressive, independent Scotland within Europe.
“The people of Scotland deserve the choice of a better future than the Brexit Britain being imposed on us.”