As San Francisco prepares to open 200 Navigation Center beds on the Embarcadero, Supervisor Matt Haney is trying — again — to force the city to open shelters in every part of the city, even those without large numbers of homeless people.

Haney wants to encourage the Department of Homelessness and Supportive Housing to open a Navigation Center within 30 months in every supervisorial district that doesn’t have one. The point of his legislation is to spread homeless services beyond neighborhoods where they are now concentrated — SoMa, the Mission and the Bayview.

City leaders have been searching for sites to add more shelter beds as the homelessness crisis has worsened. But Mayor London Breed, the Department of Homelessness and Supportive Housing, and several supervisors say they don’t know how legislation such as Haney’s could make them move any faster. His measure would include no punishment if the department doesn’t comply.

Still, Haney says the city isn’t moving with enough urgency.

“The city needs to be pushed,” he said. “City Hall is not stepping up adequately to address this as a crisis where every neighborhood is part of a solution.”

He originally introduced the idea to the Board of Supervisors in April, after Breed proposed the Navigation Center on the Embarcadero. Haney, who said he did not know about the proposal until shortly before it was made public, was heavily criticized by constituents who live near the site and didn’t want a homeless shelter in their neighborhood.

“New beds have been added at a glacial pace as the conditions on the streets have worsened and homelessness has shot up,” Haney said. “Now we’ve got no choice but to force them to do it.”

Finding a site for a Navigation Center is costly and complex. The sites need to be the right size and right price, and in a location where homeless people will go. Since the shelters are meant to be temporary, the homelessness department says spending the millions of dollars it takes either to build one from scratch or renovate an existing building becomes hard to justify.

“We have looked all over the city for places, and it doesn’t always work out,” said Jeff Kositsky, director of the homelessness department. “In a world where there are limits, both financially and geographically, we need to make sure we are using what we have as effectively as possible.”

Haney is reviving his proposal as Breed chips away at her goal of opening 1,000 shelter beds by the end of next year. Since taking office, she has opened 366 shelter beds and has 424 in the pipeline, including the 200 set to open on the Embarcadero.

Jeff Cretan, a spokesman for the mayor, said she is working on opening more beds, and the city “doesn’t need legislation and additional bureaucracy to do that.”

“But we do need available sites, funding and political will,” Cretan said in a statement. “The mayor has shown she’s willing to fight to deliver these Navigation Centers, like she did on the Embarcadero, and she will continue do just that as we continue our work to identify new locations to help people struggling on our streets.”

Under Haney’s legislation, a homeless shelter would qualify as a Navigation Center if it has at least 40 beds, establishes an individualized “care plan” for each resident’s housing and health needs, and focuses on getting people off the streets and into permanent housing. People must be allowed to stay for at least 90 days, and may stay longer “as long as they are participating in assigned services.”

The homelessness department would be asked to open two Navigation Centers within six months in supervisorial districts that don’t have one. Then it would have two years to spread the shelters to the remaining districts without them.

Facilities like the “safe” parking lot planned for near the Balboa BART station for homeless people who live out of their vehicles would qualify. But permanent supportive housing, drop-in facilities, or board-and-care homes for the elderly and mentally ill — which the city is losing at a rapid rate — would not.

Nearly all supervisors say they are open to having Navigation Centers in their districts. But only Haney and Supervisors Hillary Ronen and Shamann Walton have the shelters in their neighborhoods.

Supervisor Aaron Peskin has proposed a site at 888 Post St. in his district, and said the city is exploring the possibility. Supervisor Rafael Mandelman said the city has looked at several sites in his district, which includes the Castro, but hasn’t settled on one.

Supervisor Gordon Mar, who represents the Sunset, said he supports spreading services around the city. But he doesn’t think a Navigation Center is necessarily the best fit for his district, which has one of the lowest homeless populations in San Francisco.

Instead, he would like to be able to consider options like transitional housing or tiny houses, like those in Oakland.

“It’s important to allow each district the flexibility to decide what is the best model and strategy that fits the needs of their districts,” Mar said.

Editors note: An earlier version of this story incorrectly stated that no new Navigation Centers have been announced since the Embarcadero one. The Bayview shelter was announced after the Embarcadero.

Trisha Thadani is a San Francisco Chronicle staff writer. Email: tthadani@sfchronicle.com Twitter: @TrishaThadani