Opponents of the proposed Houston Equal Rights Ordinance have been present at early voting centers. (Pat Sullivan/AP)

Among the political signs jammed into the grass outside a polling station in this city’s South Park neighborhood stands one placard bearing an unusual slogan: “No men in women’s bathrooms.”

The statement, which is also emblazoned on T-shirts and conveyed in ominous television ads, has become a rallying cry for opponents of a measure designed to protect gay and transgender people from discrimination in Houston, the nation’s fourth-largest city and one of its most diverse.

The campaign to pass the Houston Equal Rights Ordinance, or HERO, has become a priority for national gay rights groups and the city’s gay mayor — as well as for local business leaders, who fear an economic backlash similar to the one that hammered Indiana earlier this year if it is not adopted. But with an election set for Tuesday, polls show voters are divided on the measure — and some analysts are predicting defeat.

One reason, they say, is the provocative claim that the measure would permit “any man at any time” to enter a women’s bathroom “simply by claiming to be a woman that day.” Opponents have dubbed the measure “the bathroom ordinance.”

“Houston voters do not want men in their women’s bathrooms,” said the Rev. Dave Welch, executive director of the Houston Area Pastors’ Council. “It’s an invasion of privacy, an invasion of a safe space for women and girls.”

It’s also completely untrue, supporters of the measure say. They accuse Welch and others of fearmongering, noting that the ordinance would not only protect the rights of transgender people, but also challenge discrimination on the basis of race, sex and a dozen other factors, including military status.

“I don’t care who you are, you belong in one of those 15 categories. This literally is going to protect everyone in the city,” said Lou Weaver, 45, a transgender activist who is working to pass HERO. “Trans women are literally being painted like boogeymen here, the monster under the bed.”

The clash has become this fall’s marquee battle for gay rights activists, with groups such as the Human Rights Campaign and the American Civil Liberties Union pouring hundreds of thousands of dollars into the campaign to persuade voters to enact the ordinance.

The federal government does not explicitly bar discrimination against those who are gay and transgender. To date, 17 states and more than 200 municipalities bar discrimination in housing, employment and public accommodations based on sexual orientation and gender identity, according to Freedom for All Americans, a gay rights group active in the pro-HERO campaign.

Texas is not one of those states. So last year, Houston Mayor Annise Parker, a longtime lesbian activist serving her third and final term in office, decided to fulfill a campaign promise to enact a nondiscrimination ordinance in her city.

The city council narrowly approved the measure after a contentious debate. But opponents, led by a coalition of local pastors, immediately moved to have the measure struck from the books and put on the ballot.

Parker fought back. Attorneys for the city subpoenaed sermons delivered by five pastors who had been critical of the ordinance, accusing them of illegal electioneering from the pulpit, among other things. But the move drew national criticism, and Parker withdrew the subpoenas.

After a protracted legal battle, the Texas Supreme Court ordered the city to put the ordinance on the November ballot or repeal it. The ordinance had been in effect for only three months. A referendum was set for Tuesday.

Parker has said the issue is personal for her, not only because she is gay but also because her son is black and her three daughters are mixed race. Race has emerged as a huge issue in the election: Of 11 complaints filed under HERO during the brief time it was in place, five alleged racial discrimination and one was related to gender, according to the mayor’s office.

Last month, the race issue took center stage after three black lawyers said they were asked to pay a cover charge at the Gaslamp, a popular Houston bar, where white customers were admitted free. While they are filing a federal lawsuit, the attorneys have become ardent advocates of HERO.

“In a city this diverse, with 90 languages spoken and all the varied cultural and racial and religious influences, you have to have a city where people respect each other and find mechanisms to communicate and get along with each other,” the mayor said in an interview. “Voting down HERO sends exactly the counter-message.”

Parker noted that city law prohibits people from entering bathrooms for the purpose of harassment, and suggested that the bathroom campaign is driven by antipathy. “I’ve been a lesbian activist for 40 years, since the ’70s, and these same people have been on the other side,” she said. “Now, they’re targeting the transgender community.”

Some opponents of HERO say they are not worried about transgender people, but instead about sexual predators who could pose as women to gain access to women’s restrooms. Others object to the fact that the ordinance allows people to self-identify as women; they argue that the law should set a standard, such as a public declaration or a medical procedure, to establish gender identity.

“I’m sorry. If he’s transgender, he should stay home until the process is complete,” said Loyce Johnson, 70, a retiree who is volunteering for the Campaign for Houston, which opposes HERO. “Anybody with a penis, I don’t want them in the ladies’ restroom.”

Robert M. Stein, a pollster and political science professor at Rice University, said his latest poll, which wrapped up in early October, shows the measure passing by six percentage points. But the political winds have shifted since then, Stein said, and now he is predicting defeat.

Early voting has shown a strong turnout among Republicans, as well as among African American voters, who appear less friendly to the ordinance than white Democrats, Stein said.

The anti-HERO sentiment was evident last week outside the Palm Center, an early voting location in predominantly black South Park. Many people wore “No Men in Women’s Bathrooms” T-shirts and handed out literature.

“I voted no,” said Brittany McFarland, 25, a state social worker. “I’m just not comfortable sharing a bathroom with a man.”

Others emerged from the polling station to report that they had voted for the ordinance.

“I’m for equal rights,” said Faye Sadberry, 52, a doctor. “There wasn’t anything in there about bathrooms.”

Predictions that the measure could fail have sparked panic among members of the business community, who worry about creating a perception that Houston is unfriendly to gays. Earlier this year, Indiana officials were forced to backtrack after the governor signed a measure to protect religious freedoms — and that measure was labeled anti-gay, prompting sports organizations, performers and major businesses to threaten to pull out of the state on the eve of the NCAA basketball playoffs.

Similarly, the governor of Arizona last year vetoed a bill decried as anti-gay amid fears that the National Football League would move the Super Bowl to a different state. Houston is scheduled to host the 2017 Super Bowl, a fact that was mentioned at a news conference last week hosted by the Greater Houston Partnership, the city’s chamber of commerce.

HERO “is a statement of who we are as Houstonians,” partnership President Bob Harvey told reporters. “Let’s make a positive statement to the country and the world. It is important for business.”