Black and Hispanic students make up nearly 70 percent of the city’s public school students, but they received just 10 percent of offers for seats at specialized schools this fall. Both white and Asian students, on the other hand, are overrepresented. About 27 percent of the offers went to white students, who make up 15 percent of the school system; 52 percent went to Asian students, who make up 16 percent.

“We have to make sure the very best high schools are open to every New Yorker, every kind of New Yorker,” Mr. de Blasio said. “They need to look like New York City.”

Mr. de Blasio’s plan has two parts. The first, which the city can do on its own, would set aside 20 percent of seats at each of the eight specialized schools for poor students who just miss the cutoff score on the test. Those students would be pulled from high-poverty schools, which tend to have a higher concentration of black or Hispanic students. They would be eligible to earn their seat by completing the city’s summer Discovery program.

The city has estimated modest gains from these changes, which will begin going into effect for the September 2019 admissions cycle. Officials expect the percentage of black and Latino students at those schools to rise to about 16 percent of seats from 9 percent.

The more substantial proposal calls for eliminating the test entirely and instead basing admissions on a combination of a student’s class rank and scores on state standardized tests, taking the top 7 percent of students from each of the city’s middle schools. The Education Department said that about 7 percent of specialized seats would be set aside for students from nonpublic schools, like private schools or Catholic schools. High performers from those schools would be chosen by lottery.