The SNP will hold on to 50 seats in the general election on June 8 but the Tories are expected to take six seats from Nicola Sturgeon's party, according to a new Ipsos Mori poll by STV.
SNP deputy leader Angus Robertson is also on course to lose his seat in the General Election, according to analysis of a new poll.
The survey, carried out for The Herald by BMG, puts support for the SNP at 43 per cent, 13 points above the Scottish Conservatives on 30 per cent.
It has Labour on 18 per cent, the Liberal Democrats at 5 per cent and the Scottish Greens at 2 per cent, after "don't knows" have been excluded.
According to the Electoral Calculus projector, the SNP would lose seats to the Tories.
Nicola Sturgeon told the BBC that if the Conservatives did fail to win an outright majority, her pro-independence Scottish National Party would consider supporting a "progressive alliance" of other opposition parties on a case-by-case basis.
"The most likely outcome here is a Tory (Conservative) victory, but a Tory victory no longer certain of an increased majority," she said, discussing her reading of opinion polls.
Speaking in Perth this week, Ms Sturgeon said if the SNP wins the most seats in Scotland then "continued Tory attempts" to block an independence referendum would be "democratically unsustainable".
Asked if losing seats would weaken her case, Ms Sturgeon said: “Maybe I’m old fashioned. But the party that gets the most votes and more seats than any other wins the election. That’s democracy.”
Tory Prime Minister Theresa May has blocked the SNP's plan to hold another referendum on Scottish independence after most Scots voted to remain in the EU during the referendum last year.
The SNP looks almost certain to win the most seats in Scotland in the general election after virtually wiping out Labour in Scotland in the last election in 2015.
Labour clung on to just one seat in Scotland and lost a total of 40 seats, while the SNP won 56 seats in a remarkable landslide victory. The Lib Dems and Tories have one seat each.
A recent poll of 2,000 Scottish voters has found that SNP support has fallen below 40 per cent for the first time since 2014 due to a Tory surge in Scotland.
The SNP were at 39 per cent, the Tories at 29 per cent, Labour at 25 per cent and Lib Dems at four per cent, according to the SurveyMonkey poll for the Scottish Sun on Sunday.
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Scottish polls: Who will win election in Scotland? Ruth Davidson v Nicola Sturgeon
But a YouGov poll, published in late May, put the SNP at 42 per cent, the Conservatives at 29 per cent, Labour at 19 per cent and the Liberal Democrats at nine per cent.
Professor John Curtice, who runs the What Scotland Thinks blog, said that the YouGov poll represented a one-point increase for the SNP, Tories and Labour since the end of April.
“But while this poll might indicate that little has changed during the course of this election campaign, it confirms that this election is currently heading towards a somewhat different outcome from that in 2015," he posted on his blog.
“The SNP may still dominate the electoral scene in Scotland – indeed, the party’s level of support for the current Westminster contest would have been regarded as truly remarkable before 2015 – but it looks as though the SNP are unlikely to replicate fully their astonishing success two years ago in winning 50% of the vote.
— NumbrCrunchrPolitics (@NCPoliticsUK) May 28, 2017
“Meanwhile the SNP is facing a different principal challenger this time, in the form of a revived Conservative Party rather than a declining Labour party.”
YouGov said: “Most attention in Scotland has been on the Conservative resurgence at Labour’s expense – the Tories have nearly doubled their vote share, increasing from 15% in 2015 to 28% now.
“Labour meanwhile have seen their share of the vote decline from 24% to 18% now – their lowest share of the vote anywhere in the UK.
“This is something of a sideshow to continued SNP dominance, however. The nationalists remain in command at 41% of the vote, although this does represent a nine point decrease on their share of the vote in 2015.”
This graph shows how the polls compare for the 2015 election and 2017 election
The latest UK-wide poll, by Survation for ITV's Good Morning Britain, put the Conservatives at 43 per cent – only six points ahead of Labour at 37 per cent.
Scottish Conservatives leader Ruth Davidson has dismissed the nationwide fall in support in the polls as something that "always happens" in election campaigns.
"As a veteran of six Scottish or UK wide elections and two referenda the narrowing of the polls always happens around this time and it focuses people's minds,” she told Sky News.
Ms Davidson has attacked the SNP’s domestic record in Scotland and fiercely opposes Ms Sturgeon’s plans to hold a Scottish independence referendum.
The SNP may have lost support from its most Eurosceptic voters by taking a stand against Brexit and pushing for Scotland to remain part of the EU.