The party made the claim after a new poll finding more than a third of people were more likely to back leaving the UK after the country exits the European Union.
Applying that to the result of the 2014 referendum - in which Scots voted by 55% to 45% to stay in the Union - could result in a 51.39% vote for independence, with 48.61% against, according to its analysis.
SNP depute leader Keith Brown said the figures showed a “significant swell in support for Scotland taking all decisions over our future for ourselves” following Britain’s departure from the EU.
He said: “As the chaos and despair at Westminster’s handling of Brexit continues, more and more people are drawn to the opportunities and hope for the future that independence offers.”
The research, carried out by Survation for Channel 4, has been described as the biggest piece of independent research on attitudes to Brexit, with 20,000 people across the UK questioned.
Last night, Survation responded to the SNP claim. In a tweet to SNP strategist Ross Colquhoun the organisation said: “There isn’t a headline Scotland voting intention in tonight’s tables (Scotland was not weighted separately, so calculating the Westminster crossbreak for Scotland does not get you there.
“There is also no Scottish Indy Ref voting intention.”
Overall the survey found that the UK would now vote to remain in the EU by a majority of 54% to 46%.
In Scotland, where 1,725 people were questioned, the poll found 64.7% want to stay part of the EU, with 35.3% in favour of leaving - a larger majority in favour of remain than in the 2016 referendum.
Asked how Brexit would affect their vote in an independence referendum, 37.8% of Scots said they were more likely to back leaving the UK - with just over 26% much more likely to do so.
Meanwhile 25.4% said they were more likely to vote to stay in Britain, while 30.6% said they were neither more or less likely and 6.2% did not know.
More than half (53.8%) of Scots said Brexit would be bad for the economy, while 43.8% said it would have a negative impact on the NHS.
Meanwhile 45.2% of voters north of the border fear leaving the EU will be bad for their household finances.
Mr Brown said: “People are right to be concerned about the harm that Brexit will do to our economy, our NHS and their household incomes. Brexit could cost every person in Scotland £2,300 a year and will put 80,000 jobs on the line.
“The more Tory and Labour politicians ignore Scotland’s interests and seek to stop us from making decisions over our future the more support for the SNP and independence will continue to grow.”
Clarification: A previous version of this story had the erroneous headline: ‘SNP hails new poll ‘showing slight majority’ in favour of independence’. This was incorrect and has since been changed. The Scotsman apologises for any confusion caused.