As the exiting speaker, Ms. Mark-Viverito holds few cards left with which to influence her members, apart from the power to allow them to see their own bills get a vote.

The street-vendor proposal would increase the number of vendors by 330 per year over 10 years, above the current cap of 5,100, while providing for a new office of enforcement. Only those with licenses could operate their vendor carts, a new regulation that Ms. Mark-Viverito’s office said would mean fewer than the maximum number would be on the street. (Under current law, a license holder can hire workers to staff a cart 24 hours a day.)

For Ms. Mark-Viverito, the legislation is important partly because many street vendors are immigrants, and they rely on such work for their livelihood.

“It’s complicated, and it’s considered one of these third-rail issues, nobody wants to touch it,” she said. “I feel it’s a very balanced bill.”

Mr. de Blasio supports the other contentious measure, two policing bills, known together as the Right to Know Act, that would require officers to provide information during many street interactions, including presenting a business card and giving the reason they are questioning the person, and, separately, to get consent from a person before conducting a search in cases where a search would otherwise not be permitted.

But while police reform advocates and the New York Civil Liberties Union support the search bill, they have railed against the compromise over information in street interactions reached between the Council and the Police Department, which would exclude car stops as well as low-level interactions in which officers engage in questioning without yet having suspicion that the person was involved in a crime.

When that measure was discussed at last week’s closed-door leadership meeting, Councilman Jumaane D. Williams of Brooklyn spoke up forcefully. Mr. Williams, who is a candidate to be the next speaker and has a history of shepherding police reform bills, objected to the deal in an exchange with the speaker that one person familiar with discussions called “heated.”