“We have to come up with a universal way to make sure that that opportunity is there for our young people,” Mr. de Blasio said at the time.

But the phones, which would be regulated on a school-by-school basis, can pose numerous problems. Some principals, particularly those of schools with high rates of behavioral problems, have privately said they oppose lifting the ban. They worry about the potential for cheating and the risk of theft. When fights break out, they say, students with cellphones in their pockets can summon a much larger crowd.

A principal of a high school with metal detectors, who spoke on the condition of anonymity because he was not authorized to talk publicly on the matter, said he was extremely disappointed with the decision. “This increases the potential for incidents,” he said, adding that iPhones make appealing targets for theft, which would be recorded as a major incident on a student’s record.

“Kids are going to stage fights so they can put up posts on social media,” he said.

There are also more prosaic concerns, such as students being distracted in class.

Under the new policy, principals would decide, in consultation with teachers and parents, on a range of options for cellphone use, according to The New York Daily News. They can require students to store phones in backpacks or in a designated place; allow use of phones during lunch or in designated areas; or allow phones to be used for instructional purposes. For schools that do not choose a policy, the default rule would allow phones that remain hidden, the de facto policy currently in many schools.