The fight centers on an affirmative fair housing rule put in place in 2015 after two years of study and public hearings. At the time, Julian Castro, Mr. Carson’s predecessor as HUD secretary, said the regulation would “provide all Americans with access to safe, affordable housing in communities that are rich with opportunity.”

But housing advocates say the administration has not followed through with ensuring equitable treatment in housing, and they are concerned that HUD may discriminate against vulnerable populations when awarding funds to reconstruct areas damaged by natural disasters.

Lisa Rice, president of the National Fair Housing Alliance, one of the groups bringing the suit, said the litigation was intended to prevent racial bias in the reconstruction of housing, drainage systems and other infrastructure. The group wants to ensure that black, Hispanic and low-income residents of Houston and Corpus Christi are not put at risk in the event of another natural disaster.

“This is the fight Dr. King envisioned, even 50 years later,” Ms. Rice said.

Madison Sloan, director of Texas Appleseed’s Disaster Recovery & Fair Housing Project, another plaintiff in the lawsuit, said the suit was about not leaving people behind.

“We know that disasters have a disparate impact on the most vulnerable populations, especially communities of color,” she said. “Historically, segregation has forced these communities into the areas most vulnerable to natural disasters. We have to make sure we are rebuilding in a way that remedies, rather than perpetuates, these inequalities.”