The bedraggled veteran draped in a flag, begging for spare change, has been one of the most persistent images of homelessness in this country.

In New York City, however, it is a sight that may be fading from view.

Even as city officials have struggled to cope with the surging number of homeless people over all, the city has quietly made significant inroads over the last few years in finding permanent housing for one group: homeless veterans. The number of veterans staying in shelters or on the streets has plummeted by nearly 80 percent since 2009, according to the federal Department of Housing and Urban Development.

Officials now say there are 760 homeless veterans in the city, down from 3,689 six years ago. Although some advocates for homeless people have questioned the accuracy of the tally, they acknowledge that significant progress has been made.

The executive director of the United States Interagency Council on Homelessness, which coordinates the federal response on the issue, said in a letter to Mayor Bill de Blasio this week that the city had “effectively” ended chronic homelessness among veterans.