A new study by polling firm ICM Research found 39% are backing independence, up two points from last month.
Over the same period the proportion of people wanting Scotland to stay in the UK fell from 49% to 46%.
The poll, carried out for the Scotland on Sunday newspaper, found 15% of the 1,010 people questioned were undecided about how to vote in the referendum, to be held on September 18.
More than two fifths (43%) said independence would be bad for the economy, while 36% said leaving the UK would be good for Scotland's economy, with 6% of people believing independence would make no difference to this, and 13% did not know.
More than a third (36%) said independence would mean less inequality, while 16% believe it would increase inequality, with 30% saying there would be no difference and 18% did not know.
Meanwhile a third of Scots do not believe Holyrood will get more powers in the event of a No vote in the referendum.
When asked what they thought would happen if this was the case, 39% said the Scottish Parliament would get more powers and responsibilities, while 33% said it would keep the same powers as it currently has, and 13% said they thought Holyrood would lose some of its powers, while 15% did not know.
Blair Jenkins, chief executive of the pro-independence Yes Scotland campaign, said: "'With this poll, a swing of only around four points is all it takes to put Yes ahead - which we are confident of achieving - and we'll continue to step up our campaign work in all of Scotland's communities in the months ahead.
"It is the highest Yes rating in an ICM poll since January 2012, and this poll also suggests that a majority of people in Scotland don't believe that the Scottish Parliament would get more powers if there was a No vote - that is why a Yes vote is so important.
'It's yet another poll confirming that more and more people are moving to Yes. A Panelbase poll published last week also highlights that the gap is continuing to narrow."
He added: "More people in Scotland are realising that the additional powers being talked about by the anti-independence parties are far too weak and limited, as well as unlikely to be delivered - and that the only way to achieve a Scottish Parliament with the powers to build a more fair and prosperous society is by voting Yes."
A spokesman for the pro-UK Better Together campaign said: "This poll shows that the gap between yes and no is exactly the same as it was two months ago.
"While we can take comfort from the fact that this poll, like all others, show that a clear majority of us want to remain in the UK, we need to make sure that we are fighting for every vote between now and September.
"The choice we face in the referendum is clear. Do we want to take a huge leap into the unknown with independence, or do we want to have an enhanced Scottish Parliament backed up by the strength, security and stability of the UK?"