The set-asides for affordable units would range from 20 percent to 30 percent of all the new units built, with the rest renting at market rates in the neighborhood. The decision of what level of affordability to require in each project would be determined on a case-by-case basis by the local Council member. Developers would also be able to create the units of affordable housing off-site at another location, though the plan discourages this by increasing the amount of units that would need to be affordable.

Administration officials had argued that they had designed their zoning proposals so they were financially viable for developers, and could withstand any potential legal challenges. But there were signs of impending change over the weekend: The mayor’s office announced that, after discussions with tenant advocates, it agreed to do a study that would seek to find another means — possibly a separate program — for the city to create housing for those with even lower incomes, and that would require that higher standards at construction sites to encourage union labor.

Both remain demands of those advocates, like a coalition of housing and labor groups known as Real Affordability for All, that originally opposed Mr. de Blasio’s plan. The coalition director, Maritza Silva-Farrell, described the study, whose contours and timing have yet to be fully determined, as a victory because, she said, it would focus on how to meet their demands, not whether doing so is possible.

Michelle de la Uz, an affordable housing advocate and nonprofit developer who sits on the City Planning Commission, said the changes announced on Monday were an improvement to the mayor’s plan. However, “if we’re going to be an inclusive city,” she said, the conversation must continue about how to provide housing for the poorer households still left out from the options.

Others said the changes to the city’s proposals could dampen the creation of new units of housing.

“It is great that the Council wants to get to ‘yes,’ but their design and affordability proposals add up to a more expensive, less flexible program that will likely produce fewer affordable units over all,” said Kathryn S. Wylde, the president of the Partnership for New York City, an association of the city’s business groups.

“The Council’s call for 20 percent of units to be rented at 40 percent of A.M.I. will typically require additional subsidy from another source, which limits the volume of production that can be achieved through the zoning package alone,” she added, referring to the area median income.

The mayor had also proposed numerous zoning changes that also face strong opposition because they would allow taller buildings and eliminate some parking requirements for new development, among other concerns.

The Council, to assuage critics, made a series of granular changes to the mayor’s zoning plan, including eliminating height increases in certain areas of Manhattan, and preserving more parking for developments in neighborhoods far from public transportation.