“As a city, with the work we’re doing as a Council, we’re really trying to lead the way on conversations that you’re starting to see nationally,” Ms. Mark-Viverito said in an interview.

She acknowledged that her policing agenda had made even some council members anxious, but said she planned to “stay the course” despite what she described as exaggerated criticism in the press.

“Yes, people are going to be nervous about that,” she said. “People are talking about ‘the sky is falling.’ ”

“The mainstream media, the tabloids, have pushed against this,” she added.

The burst of attention-grabbing activity from a typically subdued figure has prompted new interest in Ms. Mark-Viverito’s future. There is no natural next step for her in 2017, when she must leave the Council because of term limits. She has pledged not to run for Congress next year, and every citywide office is occupied by a Democrat who is expected to run for re-election in 2017.

Ms. Mark-Viverito has expressed interest to at least one ally in running for mayor someday; she has also made lighthearted remarks in private about possibly seeking the governorship of her native Puerto Rico, also according to confidants, who asked to speak anonymously about private conversations.

Ms. Mark-Viverito said she was in no rush to make decisions about her political future. She called the idea of running for governor of Puerto Rico “absolutely hysterical,” though she did not rule it out. Asked if she would consider seeking state or federal office in the future, she said, “I think the city is what interests me most.”