San Francisco Mayor Ed Lee has set his sights on the ambitious goal of pulling 1,000 homeless people off the streets this winter and, to kick-start that effort, he wants to open two new homeless Navigation Centers.

Getting that many people housed will take the combined labor of several city agencies, including the police, Public Health Department and Homelessness and Supportive Housing Department. But Lee told The Chronicle on Tuesday that he wants to “make a big move while it’s raining and cold.”

“I feel people’s frustrations when we’re bringing people into the Navigation Centers we have running already and they don’t seem to see things change,” Lee said. “We have to be smarter and more dedicated.

“So let’s do our best during the winter months, when the feelings are so strong about helping the homeless among us during the bad weather,” he said. “We have to translate that into action.”

The mayor is negotiating with the California Department of Transportation for two sites for Navigation Centers, city shelters that also provide intensive services. One is at Fifth and Bryant streets under the Highway 101 approach to the Bay Bridge, the other at 13th and Division streets.

The Bryant Street location is an empty lot across the street from the city’s largest homeless shelter, the Multi-Service Center South. For many years, it has frequently hosted large tent encampments. On Tuesday, there were a dozen tents and damp sleeping bags on the ground, and several people living in them said they’d happily trade their blankets for a bed in a Navigation Center.

“It’s a great idea, and a great location if people could stand the freeway noise,” said Shane Peak, 35, who slept part of the morning in the middle of the Caltrans lot. “There’s so much fighting and mess out here, it’s hard to take it. Most of us just wish we had a roof somewhere.”

The other site, on 13th Street, is now leased out by the city as a parking lot. Supervisor Hillary Ronen, who represents that area and has been vigorously looking for sites in her district to open Navigation Centers, said she likes the mayor’s proposal.

“I share the mayor’s urgency, and I’ve been ringing the alarm bell since I got into office to have more safe places of dignity for homeless people and more relief for business owners and residents,” she said. “That site would work, and I’ve got my eyes on other sites, too, that are also in locations with tons of encampments and in industrial areas.”

Right now, the city has two Navigation Centers in the Mission, one in Dogpatch, one in the Civic Center and one at San Francisco General Hospital. They are popular with both homeless people, who are allowed to move in with their partners and possessions, and with service providers. They are intended to move people from homelessness to permanent or supportive housing, and the main complaint comes from those who get only short stints and are turned back out before they can be housed.

The two Mission District buildings are scheduled to close next year, but Ronen said she has found new facilities to replace them.

Lee has not identified where funding would come from for two new centers, and there is no timeline for opening them. The state, however, allocated $10 million over the summer to San Francisco for new Navigation Centers, and most of that money is still available. Lee said he also is considering two or three additional sites for centers.

The challenge, though, is that permanent housing, which for the most troubled people means supportive housing with counselors on site, is in short supply, with only a few hundred new units being planned each year.

Lee said he anticipates more affordable housing will become available as set-asides in buildings are completed. The Moving On program, which helps people move out of supportive housing as they become able to manage on their own, will be expanding as well. And 295 temporary shelter beds are being opened in churches for the winter.

Lee additionally has high hopes for more modular “stackable” homeless housing complexes, like one being built at Seventh and Mission streets, and he is talking with unions about establishing a manufacturing facility in San Francisco, in addition to the one in Vallejo already up and running.

Lee talked up some of his plans at the annual Interfaith Thanksgiving Prayer Breakfast on Tuesday morning, and many there said they believed they are achievable. Among those was Kaushik Roy, chairman of the San Francisco Interfaith Council.

The council helped found the city’s first Navigation Center in 2015 and organized the annual winter shelter program that began opening up those 295 seasonal shelter beds Sunday. The people who occupy those beds will be counted among the 1,000 Lee wants to bring inside. Roy said his council’s housing specialists will help the mayor’s staff find affordable housing spots in the winter.

“Everyone needs to step up — government, the nonprofits, the faith community, everyone,” Roy said.

Officials in the city homelessness and police departments said they are looking through their resources to find ways to ramp up their efforts.

“It’s doable, and having the renewed focus from the mayor’s office helps,” said Randy Quezada, community relations manager at the homelessness department.

Kevin Fagan is a San Francisco Chronicle staff writer. Email: kfagan@sfchronicle.com Twitter: @kevinChron