The Panelbase survey for the Sunday Times suggests Ruth Davidson’s party could overtake Kezia Dugdale’s on May 5.
The poll puts support for Labour in the constituency vote at 19 per cent, slightly ahead of the Tories on 18 per cent. Backing for the SNP is at 51 per cent, while the Liberal Democrats are on 5 per cent.
In the regional list vote, the Conservatives edge ahead of Labour, on 19 per cent and 18 per cent respectively.
Support for the SNP is at 47 per cent with backing for the Lib Dems at 4 per cent and the Greens at 8 per cent.
The poll suggests Ms Davidson is the more popular leader, with a net approval rating of -8 compared to -17 for Ms Dugdale, although both languish far behind SNP leader Nicola Sturgeon on +16.
Scottish Liberal Democrat leader Willie Rennie has a net approval rating of -13, while Scottish Greens co-convener Patrick Harvie is on +8 per cent.
Around a third (32 per cent) of those questioned indicated support for a rise in income tax to improve Scotland’s schools, a key Labour policy.
However, 37 per cent said they opposed the idea and more than two thirds (69 per cent) thought that income tax should remain the same as in England, as proposed by the Tories.
The poll of 1,021 voters, carried out between April 6 and 15, coincides with publication of a report suggesting that a vote for the SNP on the regional list ballot might be “wasted”.
The paper by psephologist Professor John Curtice for the Electoral Reform Society (ERS) Scotland argues that the SNP is projected to do so well in constituency contests that it could win as few as two seats through the proportional representation part of the ballot.
Prof Curtice said: “That would appear to imply that under this scenario many a list vote for the SNP would be ‘wasted’, that is it would fail to contribute towards the election of an MSP.
“Indeed, under our scenario that proves to be case for any regional list vote cast for the SNP anywhere other than in the Highland and Islands region, the only region where the party is projected to win any list seats.
“That this situation could arise in a number of regions, given the SNP’s current standing in the polls, has led to speculation that nationalist supporters might be wise on the second ballot to vote tactically for a different party, such as the Greens or the left-wing RISE grouping, both of which also support independence.
“That way their vote might contribute to the election of another independence-supporting MSP rather than apparently be wasted, though this is not a strategy without risks.”