Not only will the company have to lay off its 14 Salvadoran workers, Mr. Drury said, but it was also worrying about the roughly 30 employees who are protected from deportation by virtue of a government program for immigrants who were brought to the country illegally as children. The Trump administration has announced that the program, Deferred Action for Childhood Arrivals, will expire in March.

Congress is considering creating a new program for those immigrants, perhaps in exchange for new border security spending, but no deal has been reached. On Tuesday, President Trump said he was open to a comprehensive deal that would shield not only the young immigrants, but perhaps millions of others without legal status, presumably including those with temporary protected status.

That would be a welcome development to Mr. Drury, who said he had about 40 openings. The company — which is helping to build a cancer center, the new headquarters of the mortgage giant Fannie Mae and a project at the headquarters of the National Security Agency — was already turning away work because it could not hire fast enough, he said.

“Losing people just puts us further behind,” he said.

For Stan Marek, the chief executive of Marek, a Houston-based construction company, the decisions to end temporary protections have come at the worst possible time. Houston is waiting to be rebuilt after Hurricane Harvey, yet, he said, there will be fewer people than ever to overhaul the city’s office buildings, schools, hotels and hospitals.

About 30 employees from Honduras, Haiti and El Salvador with temporary protected status have worked for him for over a decade. Some are skilled craftsmen; some are supervisors.

Mr. Marek has pushed on his workers’ behalf, even paying for a public-relations campaign to call for immigration reform.

“If they lose their status — boom, we’ll have to terminate them, and that’s not much fun, telling a guy who’s got three kids in high school, all American-born, that he’s going to be terminated,” Mr. Marek said. “They’re good people, damn good people.”