Amy Taylor, the co-legal director of Make the Road New York said the 10-year myth is pervasive, despite education efforts from her organization and others. “The No. 1 question we get from clients is, ‘do I qualify for the 10-year visa?’” Ms. Taylor said. We say, ‘there’s no such thing, why don’t we sit down and explain.’”

The promise seems real to some immigrants at first because of a tangible outcome: six months into a pending asylum application, immigrants are eligible to receive a work permit. Because asylum backlogs can last longer than two years, immigrants would not discover the fraud for a while. And under the Obama administration, they were often allowed to stay in the country even though they were subject to deportation.

For some, applying for asylum and then gambling on a deportation being overturned could be a viable legal strategy, according to Camille Mackler, the legal policy director of the New York Immigration Coalition, an activist group, But she cautioned that was only if a client had a good-faith claim for asylum, a strong fallback case for canceling the removal, and, above all, that the lawyer advised the client of the deportation risk.

E.G.R., 32, a Mexican-born mother of two U.S. citizen children, and one of the plaintiffs, said she did not realize the Hechts had applied for asylum for her until she was told to go to an interview with the government. She asked to be identified only by her initials because she is currently in deportation proceedings.

She said the asylum officer with United States Citizenship and Immigration services asked her, “Why are you asking for political asylum when you’ve been here so many years?”

She was not found eligible for asylum and has a hearing before an immigration court judge on May 18.

E.G.R. said she had been skeptical when her accountant, Mr. Guerrero, first referred her to the Hechts.