In an interview with WNBC-TV last week, former Mayor Rudolph W. Giuliani, a Republican, suggested that Mr. de Blasio’s methods were not aggressive enough.

“You chase ’em and you chase ’em and you chase ’em and you chase ’em,” he said, summing up his own approach, “and they either get the treatment that they need or you chase ’em out of the city.”

In the radio interview on Tuesday, with Brian Lehrer of WNYC, Mr. de Blasio responded. “We don’t chase human beings who are in crisis,” he said, invoking the work of Pope Francis and Cardinal Timothy M. Dolan in a plea for compassion.

Inside City Hall, appraising the magnitude of the problem has been a process. In December, a record high of more than 60,000 homeless adults and children were sleeping in shelters, according to a daily count kept by the Department of Homeless Services. As of Monday, the number had dropped to 56,731.

To gauge street homelessness, which tends to rise with the temperature, officials have pointed to a survey, conducted each winter, which in February counted 3,100 unsheltered people, slightly below the number in 2014. Through midsummer, Mr. de Blasio alluded to the count as evidence of a decline, even as advocates — long skeptical of the method — insisted the numbers were much higher, and rising.

Last month, the news website DNAInfo reported on a surge in calls about homeless people in need of help to the city’s 311 help line.