Businesses like these have boomed in China as the number of Chinese international students has steadily grown in the United States. In 2017, there were more than 363,000 Chinese students enrolled in American universities, more than a third of all international students, according to the Institute of International Education.

Jack Chen, a marketing executive for the Institute of Chinese Language and Culture, which offers college consulting and tutoring, said companies like his help students get reference letters, write essays and prepare for interviews. They also advise children on how to build up their résumés to include charity work and competitions that separate them from their peers, he said.

Mr. Chen said he knew of consulting companies that could find back doors into top universities in the United States, but he declined to disclose their names. He added that there used to be more of these services, but that American universities had cracked down after several cases of cheating by Chinese students on standardized exams and college applications.

Image Zhao Tao with President Trump and Melania Trump in a photo from 2017.

But it was perhaps the college consultants in the United States who had greater sway with Ms. Zhao’s parents, and the parents of the student in the Yale case, Sherry Guo. Both families pursued Mr. Singer’s services after meeting him through financial services companies in California, where he had formed relationships. Ms. Zhao’s family was introduced to him by an adviser at Morgan Stanley named Michael Wu, who the company said had been terminated.

Federal prosecutors have so far charged 50 people in the admissions case, in which wealthy families are accused of cheating on college entrance exams and bribing college coaches to designate students as athletic recruits. Mr. Singer has pleaded guilty to racketeering and other charges, and has cooperated with the government in gathering evidence against his clients and others he is said to have worked with.

[One scandal, two actresses take diverging paths.]

Prosecutors have not brought any charges against Ms. Zhao or her parents, nor against Ms. Guo or her parents. Both Ms. Zhao, who was a sophomore at Stanford, and Ms. Guo, who was a freshman at Yale, were expelled from their schools.