The bitter rivals are locked in a battle to lead Latin America's largest economy in a race defined by a deeply polarized Brazil.
The new survey put support for Lula at 44% in the first-round vote on Oct. 2 against 34% for Bolsonaro. A week earlier the leftist Lula was favored by 42% of those surveyed, while the far-right Bolsonaro had 34% support.
In an expected Oct. 30 run-off, Lula's lead also rose to a 10-point advantage, 50% to 40%, up by 2 percentage points from the previous survey.
The Genial/Quaest poll found that negative views of Bolsonaro's government edged up slightly to 39% from 38% last week, while the percentage of those who see the government in a positive light dipped one percentage point to 31%.
Pollster Quaest interviewed 2,000 voters in person between September 17-20. Its poll has a margin of error of 2 percentage points.