The opinion poll, by YouGov for The Times, found that support for the Conservatives in Scotland had increased by one point over the past three weeks, to 29 per cent.
This would give the Conservatives eight seats north of the border and could see them oust the SNP from Moray, Perth & North Perthshire and East Renfrewshire.
However, the poll also found that the SNP vote was holding up strongly, with Nicola Sturgeon’s party increasing its share of the vote by one point to 42 per cent.
This would give the SNP 47 Westminster seats, down nine from the 56 the party won in 2015 but still a comfortable majority in Scotland.
The poll findings come as Prime Minister Theresa May prepares to launch her party’s Scottish manifesto, a document that will promise education reforms and support for businesses while rejecting SNP demands for another independence referendum within the next two years.
On the question of the constitution, 45 per cent of poll respondents said they would back Scottish independence while 55 per cent were opposed.
The YouGov survey also indicates pro-EU attitudes north of the border are hardening, with fewer than a third of Scots now backing Brexit.
It found that 57 per cent of Scots believe that the UK was wrong to vote to leave the European Union.
This compares with 43 per cent in the whole of the UK who are still opposed to Brexit.
And, according to the poll, fewer than a third of Scots - 31 per cent - believe that the UK is right to leave the EU. This compares to 46 per cent across the UK as a whole.
If the Don’t Knows are taken out then the result is 61 to 39, similar to the June 2016 referendum result of 62 to 38 in favour of Remain.
Derek Mackay, for the SNP, said: “Now more than ever, strong SNP voices are needed to stand up for Scotland at Westminster. Labour can’t win this election in Scotland and a vote for them risks letting Tory MPs in the back door.
“Only the SNP will protect Scotland’s schools, hospitals and pensions from damaging and unnecessary Tory cuts.”
But James Kelly, for Labour, said: “In many areas all across Scotland only Labour can defeat the SNP. In seats like Coatbridge, Chryston & Bellshill, East Lothian, East Renfrewshire and others Labour is in second place to the Nationalists.
“That means Conservatives, Lib Dems and independents who have never backed us before must vote Labour on 8 June to send a message to Nicola Sturgeon that Scotland doesn’t want another divisive independence referendum.”