The rollout of the prekindergarten program has gone relatively smoothly, although some parents have complained that there are not enough seats available in certain neighborhoods, like the Upper East Side of Manhattan.

Experts have generally said the program is of good quality, considering that the initiative is still new. The city regularly evaluates its preschool providers using a nationally recognized scale for early childhood programs. The average scores from the city’s first two years were significantly below the average for classrooms in New Jersey’s Abbott Preschool Program, a highly regarded preschool effort. But they were similar to those of Abbott classrooms in that program’s early years.

While close to 70,000 children are enrolled in the city’s program for 4-year-olds, there are currently just 11,000 publicly financed seats for 3-year-olds in early childhood centers, and they are available only to families earning less than 200 percent of the federal poverty level, or $49,200 for a family of four.

In contrast to the program for 4-year-olds, which tripled the number of prekindergarten seats in two years, the new plan would be rolled out over four years. The program would start in two of the city’s lowest-income school districts, District 7 in the Bronx and District 23 in Brooklyn, and offer a seat to every 3-year-old in those districts by the fall of 2018. The city would expand the program to an additional six districts, still to be chosen, by the fall of 2020 and to all districts by 2021.

The mayor said that the city was putting $36 million in next year’s budget to finance the program and that it planned to spend $177 million annually on the program at full scale, on top of the $200 million it currently spends on seats for 3-year-olds from low-income families. But the program will cost much more than that — more than $1 billion — and Mr. de Blasio said he expected the state and federal governments to step up to help.

“We will build a coalition to put together whatever state and federal resources we need to bring this to full fruition,” the mayor said.

Whether those resources will be forthcoming is unclear. Betsy DeVos, the education secretary, has said little about early childhood education, and it was not a focus of hers as a philanthropist.