The survey, published in the Sunday Mail, found 54% plan to vote No in the independence referendum on September 18.
Just over a third (34%) of those questioned said they will vote Yes, while 12% are undecided.
It follows a separate survey for Channel 4 News published on May 1 which found 51% of people questioned were planning to vote No, compared to 37% who said they would back independence and 12% who were undecided.
The Sunday Mail survey was carried out by Progressive Partnership, which questioned more than 1,300 Scots on Wednesday and Thursday last week.
Of those surveyed, almost nine in 10 (88%) said they would vote.
The Sunday Mail reported that a Progressive Partnership poll carried out in November last year showed 27% supported a Yes vote while 56% supported No.
Better Together leader Alistair Darling said: "I am not complacent but I am not surprised by this poll. People are looking more closely at the consequences of separation - for their families, their jobs and the future of their children.
"I passionately believe that Scotland, my country and its people are better off staying in the UK. We can have the best of both worlds, with a strong Scottish Parliament with more powers, backed up by the strength, security and stability of being a partner in something bigger - the UK.
"Separation risks our future. For what? Alex Salmond refuses to tell us what would replace the pound, and can't tell us how pensions would be paid.
"It is realistic, not scaremongering, to expect answers to hard-headed questions of those who want to break up the UK."
Yes Scotland said the polls show that the trend is towards a Yes vote.
Commenting on the Sunday Mail poll, its chief executive Blair Jenkins said: "Compared with the last published survey carried out by this company in November 2013, this poll shows a significant 7% rise in support for Yes. In contrast, support for No has slipped back by 2%, narrowing the gap by 9%.
"Looking across all polls, average support for Yes has risen in every month since last November while average support for No has fallen in every one of those months, clearly demonstrating that momentum is very much with Yes.
"As polling expert Professor John Curtice points out, one thing the polling companies can agree on is that the referendum race is narrowing, and this poll confirms that.
"The other very significant figure in this poll is the 88% who say they will definitely vote in the independence referendum. With more than four months to go, this is a very encouraging indication of the high level of engagement among voters and follows recent research by ScotCen which showed that Yes supporters are more likely to vote than those on the No side.
"Different polls produce different figures, but what is clear is that the trend is very much towards Yes and we are confident that as more and more people learn about the benefits of taking Scotland's future into Scotland's hands we will achieve our goal on September 18."