But his companies have asked the federal government for permission to bring in more than 1,000 foreign workers on guest visas, according to Labor Department data. More than 200 of them were fashion models for Trump Model Management who qualify by possessing only “distinguished merit and ability,” unlike other H-1B applicants who must possess “highly specialized knowledge.”

The issue has been further muddied by questions about the immigration status of his wife, Melania, when she arrived for a nude modeling session in 1995, a year before she says she was legally able to work the United States. Ms. Trump has said she never violated the nation’s immigration laws. She also said she might not have been paid for the nude modeling shoot, but her legal pathway from tiny Slovenia to the runways of New York remains murky.

Such questions have reinvigorated opponents of the H-1B visas for fashion models, shining a fresh light on the people who shaped it and used it — including Mr. Trump; Senator Edward M. Kennedy, who sponsored legislation to add fashion models to the H-1B program; and the disgraced former New York congressman Anthony Weiner, who tried to reshape it. Mr. Weiner’s wife, Huma Abedin, is a senior aide to Hillary Clinton.

The Trump campaign did not respond to requests for comment.

Legal authorization for a model is easier to attain than it is for most professionals. United States Citizenship and Immigration Services declined to say how government screeners evaluated applicants who were models on their “distinguished merit and ability.” Angelo A. Paparelli, an immigration lawyer in Los Angeles, said, “As the Supreme Court said: It’s like porn. You know it when you see it.”