A total of 41% of people said they support Scotland leaving the UK in the latest poll by Survation - up from 39% last month.
But support for a No vote also increased by two points, going from 44% in June to 46% in the July survey.
Of the 1,013 Scots who were questioned for the research, 13% said they do not know how they are going to vote on September 18.
When these undecided voters are factored out, the survey put support for staying in the UK at 53%, with 47% backing independence.
The research also found that one in five (20.9%) of those questioned have fallen out with a family member, friend or workmate as a result of the referendum debate.
Polling expert Professor John Curtice, of Strathclyde University, said the research seems to show "the Yes side did indeed make some further progress".
But in a piece on the What Scotland Thinks website, he added: "With little more than two months to go to polling day, even a pollster such as Survation that paints a relatively optimistic picture for the Yes side still puts No ahead. That suggests Yes need to make more progress soon if victory is to look at all like a realistic prospect."
Blair Jenkins, chief executive of the pro-independence Yes Scotland campaign, described the latest poll as "extremely encouraging".
He said: "'This is the highest level of support for Yes so far in the Survation polls, and is extremely encouraging - only a 3% swing is required to produce a Yes majority in September.
"It confirms that the Yes message is getting through - that Scotland is an exceptionally wealthy country, more prosperous per head than the UK, France and Japan - but that only with independence can we make our wealth work for all the people."
Mr Jenkins added: "In line with Yes Scotland's calls for a courteous and respectful debate, it's heartening that four out of five people say they have had no arguments with family or friends over referendum issues."
Blair McDougall, campaign director of the pro-UK Better Together group, said: "This is the biggest decision we will ever take as a country so it's no surprise that passions are running high. The choice we face is clear - we can keep our family of nations together by staying in the UK or we could go it alone by turning friends, neighbours and relatives living elsewhere in the UK into foreigners.
"This is the third poll in recent weeks to show an increase in support for Scotland staying in the UK. As we get closer to the vote, Scots are thinking seriously about the consequences of separation for our pound, pensions and public services. There is a better way than separation and that's the case we will be making right up until the polls close on September 18.
"As part of the UK we can have the best of both worlds for Scotland - a strong Scottish Parliament, with the guarantee of more powers for Scotland, backed up by the strength, security and stability of being part of the larger UK. We should say no thanks to putting that at risk in September."