By the time Ms. Chicago arrived in Belen in the 1990s, New Mexico had solidified itself as a relatively welcoming place to settle for artists and writers from around the world. Many moved to small towns and villages, lured by the low cost of living and a sense of cultural diversity in a state with large Hispanic and Native American populations.

In a telephone interview, Ms. Chicago said she chose Belen partly because of the town’s location, near Albuquerque but far from big cities like New York and Los Angeles. She said this gave her and her husband, the photographer Donald Woodman, a chance to focus on their work without many distractions.

The idea for the museum was hatched by Ronnie Torres, a former mayor of Belen who also happens to be Ms. Chicago’s hairdresser and a current member of the City Council. Mr. Torres said he wanted to highlight how Ms. Chicago and Mr. Woodman had contributed to Belen, arriving with little money but eventually renovating a derelict hotel near the rail yards that is now their home.

The proposed museum would be more threadbare than opulent. Mr. Torres asked Ms. Chicago and Mr. Woodman to contribute a modest building to house the museum in addition to paying for utilities, a curator, insurance and a gift shop. Mr. Torres also asked the city to pay for a part-time employee at the museum at an annual cost of about $13,650 a year.

Some in Belen welcomed the idea, pointing to how the minimalist artist Donald Judd made the remote town of Marfa in West Texas a destination for art lovers. But in recent public meetings, members of the City Council expressed fear over how a Judy Chicago museum and the controversy it might create could change the town.

“I don’t want protesters with pitchforks and torches,” said Donald Carter, a council member since 2012. “Some of the art might upset the masses once they start looking up some of this stuff.”