The YouGov poll found Scots would vote to remain in the UK by 53% to 47% if another referendum on the issue were held now.
The polling company said the results represent a move towards the independence option of just 1% since it last asked the question in early May.
Almost two-thirds (62%) of Scots who took part in last month’s referendum on European Union (EU) membership voted to Remain.
Despite that, the UK as a whole voted in favour of Leave, prompting SNP leader and Scottish First Minister Nicola Sturgeon to warn the result makes another vote on Scottish independence “highly likely”.
For the latest research, YouGov polled 1,006 Scottish adults between July 20-25, roughly one month on from the referendum.
It concluded that a guarantee of an independent Scotland being able to remain in the EU does not move public opinion in favour of independence.
Some 46% of Scots said they would rather live in a Scotland that was still part of the UK after it has left the EU, against 37% who would rather live in an independent Scotland that remained in the EU. These numbers become 55% to 45% once “don’t knows” are removed.
The percentage who said they would not vote or did not know increased from 12% to 14% in the aftermath of the Brexit vote, the poll found.
“One month after the UK’s shock decision to leave the EU, the latest YouGov research in Scotland shows no real shift towards independence,” YouGov’s Matthew Smith said on the company’s website.
But he said the situation was a “complicated” one and added: “While Scottish independence has been mooted in the wake of the vote to leave the EU, a new referendum is not currently on the table.
“However, a lot could still change on this front in the coming years. Article 50 has not yet been triggered and once details of the Brexit deal emerge it may alter the context of the independence debate.”
The findings follow another recent poll which found almost a third of voters want to wait until the UK’s deal with the EU is clear before deciding whether to hold another independence referendum.
The YouGov survey for the pro-union Scotland in Union campaign group found 32% of Scots think the country should wait for the Brexit deal before a repeat of the 2014 ballot while 25% do not want another vote on independence until at least 2030.
Responding to the most recent poll, the SNP hailed the increase in support for independence.
SNP business convener Derek Mackay MSP said: “This poll is the latest to demonstrate that many No voters are reconsidering their opposition to independence now that Scotland faces being dragged out of the EU against our will.
“In light of the overwhelming vote to remain in the EU, it is right that the Scottish Government explores every option to protect our relationship with and place in the EU - including the option of another independence referendum if that is what it takes.
“The UK that Scotland voted to remain part of in 2014 is changing fundamentally.”
UK Government minister Andrew Dunlop said that another “divisive constitutional debate” is not want the country wants.
He said: “The arguments for Scotland remaining a part of the UK are just as compelling as they were in 2014 - in or out of the EU.
“The Prime Minister has been very clear that we are going to make a success of Brexit, and the focus now needs to be on collaborative working with the Scottish Government as ‘Team UK’ to ensure the best possible deal for Scotland and the rest of the UK.”