The Panelbase survey for The Sunday Times Scotland found that Scots are currently 53% to 47% in favour of staying in the UK.
However, backing for Scottish independence appears to grow in the event of the UK leaving the EU without a deal.
In that scenario, support for independence rises to 52% and opposition falls to 48%, the poll found.
Meanwhile, a majority (59% to 41%) said they believe independence would be better for Scotland than a no-deal Brexit.
Panelbase surveyed 1,018 voters in Scotland from April 18-24.
The poll was published on the second day of the SNP conference in Edinburgh, where party leader Nicola Sturgeon is expected to declare it is “time for Scotland to become independent”.
It comes just days after Ms Sturgeon, the First Minister of Scotland, told MSPs there should be another vote on Scotland’s place in the UK before the next Holyrood elections in May 2021.
On Saturday, a YouGov poll for The Times Scotland also recorded growing support for independence, with 49% of Scots preparing to vote Yes, compared with 45% in YouGov’s poll for the newspaper last summer.