The party, led by top candidate Elif Eralp, was on 22% in the Forsa survey for news portal t-online, followed by Chancellor Friedrich Merz's conservative Christian Democratic Union (CDU) and the far-right Alternative for Germany (AfD) on 18% each, the DPA reported.
The Greens were on 16% and the centre-left Social Democrats (SPD) on 12%.
The Berlin state election is scheduled for September 20. The city is currently governed by a coalition of the CDU and SPD.
Recent polls have shown a close race, with the CDU, The Left and AfD all leading in different surveys.
An Insa poll for the Bild newspaper published a week ago put the CDU on 20% and The Left on 19%.
The latest poll suggests no two-party coalition would have a majority. Possible options include a coalition of The Left, Greens and SPD, which t-online said was the most popular combination among respondents, or a coalition of the CDU, Greens and SPD.
Forsa surveyed 1,535 people between August 4 and 18, according to t-online. The poll has a margin of error of plus or minus three percentage points.