About a dozen mostly Democratic-leaning states including California, New York and Michigan, have refused to sign on to the administration’s directive. The nation’s two largest teachers’ unions, along with the N.A.A.C.P., are challenging the demand in federal court.

Arguments in one of those cases will be heard in New Hampshire on Thursday, escalating an increasingly tense standoff over the federal government’s role in local education.

The Trump administration is relying on a novel interpretation of civil rights law, arguing that the Supreme Court’s decision in 2023 overturning affirmative action in college admissions also applies to K-12 public schools. Federal officials say the ruling “sets forth a framework” for the use of race in education generally. And they say it requires banning curriculum and programs that are targeted toward specific racial groups, or that center on concepts such as structural racism, the idea that racial discrimination is pervasive in the economy, law and other institutions.

But that interpretation of federal law is contested by many education officials and legal scholars.

“The Trump administration is trying to use a relatively narrow decision and turn it into a broad holding that brings about whatever it wishes,” said Justin Driver, a professor at Yale Law School and an expert on the Constitution and education.

Now, several court cases are teed up to determine if Washington can withhold billions of federal dollars for schools to educate low-income students, a program known as Title I. Many of the nation’s poorest school districts rely heavily on the funds, but even affluent districts receive some Title I money if they serve low-income students.

The pressure on K-12 public education has intensified as Mr. Trump has made clear that he intends to follow through on his threats.

Last week, the administration moved to withhold federal education aid and school-meal funding from the entire state of Maine, in response to its policies on transgender athletes. It has threatened to do the same to California, because of its policies around parental notification for transgender students.

In Democratic-leaning states that oppose the move, state officials have argued they are already in compliance with federal civil rights laws.

Michael F. Rice, the superintendent of public education in Michigan, said that elite college admissions was “by definition zero sum. If you get in, I have a lesser chance of getting in.”

By contrast, he said, many D.E.I. initiatives in Michigan are “positive sum.” By expanding literature to include diverse viewpoints or creating new pipelines for teachers that also diversify the work force, he said, “you have not disadvantaged anyone.”

Chris Reykdal, the superintendent in Washington State, vowed to go to court if needed to defend state and local control over education. In a letter responding to the Trump administration, he wrote that diversity, equity and inclusion were “core values” in Washington State’s education system and that “we will not suppress or cede that to the federal government.”

“I’m not hiding this in order to keep federal money,” he said in an interview. “I’m saying, it’s what makes us successful and we should all celebrate and be more vocal.”

Hot-button issues around race and gender are core to this dispute, but the legal challenges may be decided on more routine procedural questions.

The New Hampshire case, brought by the National Education Association and the American Civil Liberties Union, argues that the Trump administration is violating congressional regulations that say federal agencies cannot dictate matters of local curriculum or instruction.

“This case is really about some fundamental failures of process at the Department of Education,” said Sarah Hinger, a lawyer with the A.C.L.U.

A similar challenge, brought by the American Federation of Teachers and other groups, is pending in Maryland. The Trump administration is expected to appeal any ruling against it, and these questions could eventually reach the Supreme Court.

The administration has argued that diversity programs violate federal civil rights law, which prohibits discrimination on the basis of race, color and national origin. It has not offered a detailed definition of D.E.I., but has given hints of some programs that it might prohibit.

For example, the administration has said programs that separate students by race, in order to provide targeted academic or social support, are a form of illegal segregation. It has also argued that lessons on concepts like white privilege are discriminatory toward white students, and that efforts to recruit more nonwhite teachers constitute illegal affirmative action.

In a statement, Madi Biedermann, a spokeswoman for the Education Department, said, “The Trump administration will no longer allow taxpayer dollars to sponsor discrimination against students.”

Title I dollars were withheld at least once before, in the 1960s, as a tool to compel school districts to desegregate.

At that time, legal experts say, the federal government was enforcing the Supreme Court’s landmark decision in Brown v. Board of Education from a decade earlier. In 1966, shortly after Title I dollars were first allocated and when enforcement of the policy was at its most vigorous, researchers estimate that about 20 percent of districts in formerly Confederate states had their Title I dollars withheld or deferred.

“There is no question they can impose fiscal penalties on state and local governments that violate the law,” said David A. Super, a professor at Georgetown Law who has studied administrative law and the federal budget.

But the government must cite a clear violation of existing law, something he says the Trump administration has not yet done.

The administration may also face another legal hurdle, because federal dollars for K-12 schools are allotted by Congress.

The Impoundment Control Act of 1974 limits the president’s authority to freeze funds appropriated by Congress. Mr. Trump has said that he wants the Supreme Court to strike down that law, giving him greater power over federal spending in a variety of arenas.

Many Republican-led states already have laws banning or limiting D.E.I. in schools, and some officials in those states have agreed to the Trump administration’s demands as a matter of course.

Texas, for example, is asking districts to sign onto the federal government’s new diversity directives by the April 23 deadline, noting that it “reinforces” existing policies in the state.

In Florida, where Gov. Ron DeSantis set an early model for Mr. Trump’s education policies, school districts said they expected few changes. “Here in Florida, there’s no anticipated impact,” said Keyla Concepción, a spokeswoman for Broward County Public Schools.

Nationwide, public schools receive only about 10 percent of their funding from the federal government — much less than many colleges, which rely on federal research grants and tuition aid.

But for many districts, like Los Angeles, the loss of those funds would still be a significant blow. Alberto Carvalho, the superintendent of Los Angeles’s public schools, said his district received more than $1 billion in federal funds annually, which support teachers’ aides, free meals and mental health counselors.

He noted that his district was not opposed to making changes in response to federal scrutiny.

But in this case, he said, Mr. Trump’s directives contradict California state regulations on how schools should handle race and gender issues.

Mr. Carvalho added that he had not been surprised to see K-12 leaders across the country rise up to resist the president, given the vulnerability of many of the children enrolled in public education.

“We are morally compelled and legally required,” he said, “to protect their rights.”