Mr. Reyes had worked up from a dishwasher to grilling steaks and preparing a la carte orders, he said, and acquired many cooking skills along the way.

The firings at the Pine Hill golf club were the result of a broader audit that the Trump Organization is conducting at its properties across the country, according to people briefed on the review. Although the organization has conducted similar reviews in prior years, this is the largest effort yet to ensure that the company’s employees are authorized to work in the country.

The review has presented yet another legal and public relations headache for company executives who are trying to contain the fallout from the revelations. It comes as the Trump Organization is facing wider scrutiny from congressional Democrats examining the company’s business practices.

At this point in the review, the issues appear to have been largely concentrated in the Bedminster and Westchester golf clubs, the people said, though the discovery of undocumented workers at the southern New Jersey property show that the matter is more widespread than initially believed. The terminations at the Westchester club were first reported by The Washington Post.

The Trump Organization announced on Tuesday that it was using a system — known as E-verify — to prevent undocumented workers from getting jobs at its properties. For the last several years, the company had utilized that system in many of its hotel properties, though it did not adopt it at most of the golf courses.

“I must say, for me personally, this whole thing is truly heartbreaking,” Eric Trump, an executive vice president with the Trump Organization, said in a statement on Tuesday. “Our employees are like family, but when presented with fake documents, an employer has little choice.”

He said that hiring undocumented immigrants was not a problem unique to the Trump Organization, and that it “demonstrates that our immigration system is severely broken and needs to be fixed immediately.”