Former New York Comptroller Bill Thompson tops Weiner with 21 percent and New York City Public Advocate Bill de Blasio holds 20 percent support.
The poll's findings follow revelations earlier in the month that Weiner, who had previously been the frontrunner in the race, had continued holding sexually explicit conversations with Internet users in 2012, after he resigned from Congress following similar behavior.
The poll found that 53 percent of likely voters think Weiner should drop out of the race while 40 percent think he should stay in.
Weiner has not indicated any plans to drop out of the race, even as his opponents and national politicians denounce him. Last week House Minority Leader Nancy Pelosi (D-Calif.) called his behavior "reprehensible."
The poll was conducted from July 24-28 among 446 likely Democratic primary voters. It had a margin of error of plus or minus 4.6 percentage points.