A Panelbase poll of 1,028 voters found 14% of those who were likely to vote planned to give their first preference to Kezia Dugdale’s party in May when “don’t knows” are excluded.
Support for the SNP was at 47%, the Scottish Conservatives were on 26%, the Liberal Democrats 5%, the Greens 4% and Ukip 3%.
Only about half (53%) of those who voted for Labour in last year’s Holyrood election said they intended to give the party first preference under the single transferable vote (STV) system used to elect Scottish councils, with 21% opting for the Tories and 19% for the SNP instead.
In the 2012 local council elections the SNP narrowly won the largest share of the vote at 32%, followed by Labour on 31%.
The Tories were in third place on 13%, 12% voted for independent candidates, support for the Liberal Democrats was at 7% and the Greens were backed by 2%.
A Labour spokesman said: “In May, voters across Scotland face a choice between electing Labour candidates who will stand against SNP cuts to local services and will oppose a second independence referendum, or SNP candidates who will seek to divide communities.
“Labour believes that together we’re stronger.”