The survey for The Herald by BMG Research suggests the SNP is set to win 68 seats while other projections show the Alba Party will take two seats, once the regional distribution of its support is taken into account.
Meanwhile, the poll suggests the Greens will pick up nine seats, meaning Holyrood would have 79 pro-independence MSPs out of 129.
Robert Struthers, head of polling at BMG, told the newspaper: “There is no question that the SNP will be returned as the largest party in Holyrood next week, but their prospect of a majority remains on a knife-edge.
“With little movement since our last poll in mid-March, there is no real evidence that any party has gathered significant momentum ahead of voters casting their ballots next week.
“Using a uniform seat calculator – a general guide of estimating how votes might translate into seats – our numbers suggest that the SNP could win a small majority of seven, thanks to a close to clean sweep of constituencies.”
BMG polled 1,023 Scots aged 16 or over between April 27 and 30.
Polling expert Sir John Curtice has claimed recently he believed there was less than a one in three chance of an SNP majority.
The BMG poll found the SNP will take a 28-point lead in the Holyrood constituency vote on 49 per cent, Labour on 21 per cent, the Scottish Conservatives on 19 per cent, and the Liberal Democrats on 9 per cent.
The survey said the SNP is on 37 per cent on the regional list, the Tories are on 22 per cent, Labour on 17 per cent, the Lib Dems on 8 per cent, the Greens on 9 per cent, Alba on 4 per cent, and Reform UK on 1 per cent.
The company’s previous survey for The Herald, carried out between March 16 and 19, found the SNP would win 66 seats, the Tories 27, Labour 20, the Greens eight, and the Lib Dems eight.
A separate poll for Scotland on Sunday found 34 per cent of voters believe the SNP has handled education well, with 39 per cent believing it was handled badly.
Alba Party leader Alex Salmond said: “An SNP regional list vote is a wasted vote across Scotland and they are set to win zero list seats on Thursday.
“It is now vital to ensure a big super-majority, with independence supporters giving the SNP their constituency vote, but backing Alba on the list.
“Today’s polls show that the independence-supporting MSPs could potentially reach 80 or even more.”
Scottish Conservative candidate Annie Wells said: “The fact is that a peach ballot for Labour is a wasted vote if you want to prevent an SNP majority and stop indyref2.
“They have no hope of challenging the SNP on the regional list.
“So I’m asking everyone like me who grew up supporting Labour to lend the Scottish Conservatives their peach ballots.”