The survey, conducted by MassINC Polling Group, shows 46 percent of East Boston voters against the project and 42 percent in support. Twelve percent are undecided.
"It's very close," said Steve Koczela, president of the MassINC Polling Group. "I'd say, big picture, this thing could really go either way."
The poll of 302 likely voters was conducted October 26-28, just a week after Suffolk Downs asked partner Caesars Entertainment to withdraw from the project.
State investigators raised a series of red flags about the gambling giant. Among them: a Caesars subsidiary signed a licensing agreement with a hotel firm that had ties to Russian mobsters.
It's unclear which casino operator will take Caesars' place.
Mayor Thomas M. Menino sought, unsuccessfully, to delay the Nov. 5 vote on the casino after the late-breaking Caesars development.
But the WBUR poll found East Boston voters not overly concerned about voting on a project in flux.
Sixty-nine percent say the election should proceed Nov. 5, as scheduled, compared to 25 percent who say it should be delayed. Voters say, by a similar proportion, that a Nov. 5 election will be "fair."
The description of the project appearing on the ballot mentions Caesars, even though the firm is no longer a part of the plan.
Koczela, the pollster, acknowledged the difficulties of surveying such a small electorate. And there was at least one unusual finding: the poll suggests men oppose the casino in larger numbers than women, a reversal of the usual gender gap on gambling questions.
But the survey points to other, more typical cleavages. Younger, wealthier and better educated East Boston voters are more likely to oppose the casino. Older, more blue-collar voters tend to support it.
Mary Catino, 83, a retired Sears Roebuck saleswoman who participated in the poll, favors a casino on the site of the horse track.
Her late husband worked as a part-time teller at the track and she's got fond memories of the place in its heyday.
"People have been gambling up there since 1930," she said. "No problem! I don't see anything wrong with it."
Joanne Oliveira, 55, a nurse who responded to the poll, said it's a tough decision. She owned horses at Suffolk Downs in the 1990s and said she'd hate to see the place wither away if voters reject the casino.
But she's leaning against the proposal. She said voters can't make a fully informed decision without a casino operator on board. And she doesn't buy the promises of high-paying jobs.
"A lot of people that are going to vote for it hear the, 'Oh, the jobs and, you know, the starting pay of $44,000 a year,' and everybody thinks they're going to be a pit boss when they're actually going to be serving drinks and cleaning toilets," she said.
Suffolk Downs is competing with two other projects for the sole Greater Boston casino license: a Wynn Resorts proposal in nearby Everett and a Foxwoods project in Milford.
The racetrack, which straddles the Boston-Revere line, has to win referendum in both East Boston and Revere on Tuesday to remain in the competition.
Campaign finance records show Suffolk Downs has spent almost $2 million over the last two years on a sophisticated, bilingual advocacy campaign, including canvassing, social media and a commercial featuring chief operating officer Chip Tuttle.
The opposition group, No Eastie Casino, had spent just $22,000 as of early this week. But Jeffrey Berry, a political science professor at Tufts University, said the group has been plucky.
"They have a lot of authenticity as well as passion," he said. "And I think they've persuaded a good number of their own neighbors that this isn't a real good idea."
It appears that local concerns about congestion and the like are paramount. Just 12 percent of voters in the poll said they are opposed to gambling on principle. Thirty-three percent said gambling elsewhere is acceptable, just not in East Boston.
Fully half said gambling is acceptable anywhere it is legally allowed.
This year's mayoral race, the most competitive in a generation, has overshadowed the casino debate in most corners of the city. But East Boston voters are tuned into the casino battle.
Seventy-two percent of poll respondents say they have been following news of the casino proposal "very closely" and 21 percent "somewhat closely." Just six percent of likely voters say they have been following the news "not too closely" or "not at all closely."
The poll has a margin of error of 5.6 percent.
This program aired on October 31, 2013.