That marks a turnaround for the disgraced former lawmaker who trailed Quinn, the longtime favorite, by 24 to 19 percent in the same poll in May.
His reentry into political life faced a rocky start, with many of his former colleagues declining to endorse his mayoral bid.
The poll also found New York City Comptroller Bill Thompson holding 13 percent support, with other candidates registering single digits.
If no candidate earns 40 percent in the Sept. 10 primary, Quinn and Weiner will likely enter a runoff.
In a head-to-head matchup though Quinn holds the edge, with 44 to 42 percent support over Weiner, with 14 percent undecided. But those figures also represent a boost for Weiner who trailed Quinn 48 to 33 last month.
More New York voters also say they would consider voting for Weiner. Forty-nine are open to backing the former lawmaker, with 45 percent saying they are not. Two months ago, 40 percent said they would consider voting for him to 52 percent who were opposed.
The poll was conducted from June 17 to 21 and has a 4-point margin of error.