A Guardian and Scotsman ICM poll shows support for Labour in the constituency votes has increased by five points since the beginning of April to 34 per cent, while the SNP has climbed two points to 34 per cent.
When the second regional choice vote, which is given to Scottish voters as part of the country's semi-proportional electoral system, was polled, the SNP lead went down to one point.
In separate a Populus poll for the Times, support for independence has fallen from 22 per cent to 21 per cent and the SNP currently have 33 per cent voter support, compared to Labour's 29 per cent.
The Liberal Democrats are positioned at 15 per cent in the Populus poll and 16 per cent in the ICM survey, while the Conservatives trail with an unchanged 13 per cent in both polls.
A loss of seats for the Lib Dems could mean a third party would be needed to form a coalition and there is currently a deadlock in talks between the Lib Dems and the SNP over any possible coalition, as nationalist leader Alex Salmon has ruled out not pushing an independence agenda.
Labour and the Lib Dems have ruled in a coalition government since 1999 and the Populus poll suggests that this will either continue or the Lib Dems will form a coalition with the SNP.
Voter turnout will be crucial for both front-running parties as the SNP believes it can mobilise as much as 60 per cent of its support.
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