The TNS poll of 1008 adults aged older than 18 found more than half (51 per cent) said they would vote to remain on June 23, up three points.
Support for leaving remained steady at 21 per cent while the remainder said they were not sure which way they would vote.
Those who backed Labour in the Holyrood election were most likely to support remaining at 72 per cent, while Conservative supporters were least likely (49 per cent).
Fifty-one per cent of SNP voters said they would vote to remain.
When those who are unsure which way to vote are removed, the poll suggests 71 per cent would vote to remain and 29 per cent would vote to leave.
Turnout looks likely to be high, with about seven in ten (71 per cent) stating they are certain to vote in the referendum, a slight drop from 75 per cent last month.
Of those certain to vote, more than half (53 per cent) back staying in, almost a quarter (24 per cent) would vote to leave and the remainder are undecided.
Tom Costley, head of TNS Scotland, said: “The level of support for the EU in Scotland has shown little change over the last few months and it looks likely that Scotland will vote to remain on June 23.
“There is still the potential for a high turnout, suggesting the Scottish public can see the importance of this decision.
“Both sides will be keen to get as many people as possible to cast their vote on the day. Given how close the race is looking across the UK, every vote will count.”
SNP leader and First Minister Nicola Sturgeon has said a vote to leave the EU against Scotland’s will could trigger a second independence referendum.
The survey, carried out between May 4 and 22, found that in a scenario in which Scotland voted to remain but the UK as a whole backed leave, 46 per cent would not support another independence referendum, compared with 43 per cent who would.
Almost nine out of ten of independence supporters would back the move, compared with only 13 per cent of those who would vote No.
If a new independence referendum was held, 38 per cent said they would back a Yes vote, 48 per cent said they would vote No. When those who are uncertain are removed, 44 per cent would vote Yes and 56 per cent would vote No, similar to the 2014 result of 45 per cent and 55 per cent respectively.