Teacher Rosemary Genuario uses an iPad to work and communicate with students diagnosed with autism at Belle View Elementary in Alexandria, Virginia, in 2012. 

WASHINGTON — School superintendents across the country are raising alarms about the possibility that Republican health care legislation would curtail billions of dollars in annual funding they count on to help students with disabilities and poor children.

For the past three decades, Medicaid has helped pay for services and equipment that schools provide to special-education students, as well as school-based health screening and treatment for children from low-income families. Now, educators from rural red states to the blue coasts are warning that the GOP push to shrink Medicaid spending will strip schools of what a national superintendents association estimates at up to $4 billion per year.

That money pays for nurses, social workers, physical, occupational and speech therapists and medical equipment like walkers and wheelchairs. It also pays for preventive and comprehensive health services for poor children, including immunizations, screening for hearing and vision problems and management of chronic conditions like asthma and diabetes.

Many school districts, already squeezed by shrinking state education budgets, say that to fill the hole they anticipate would be left by the Republican push to restructure Medicaid, they would either have to cut those services or downsize general education programs that serve all students.

“We’d have to make a local decision about what services we continue to provide and which we don’t,” said Paul Gausman, superintendent of a school district of 15,000 students in Sioux City, Iowa, that receives about $3 million in Medicaid reimbursements each year.

“I haven’t met many people who enjoy writing a check for their taxes, and I understand that,” Gausman said. “But it does not mean taxation is evil, and we’ve got to consider the most vulnerable of our population.”

Schools have been able to register as Medicaid providers and seek reimbursement, as doctors and hospitals do, since 1988. Two-thirds of districts that bill Medicaid use the money to pay the salaries of employees who work directly with children, such as school nurses and therapists.

But the Republican push to overhaul healthcare would implement a new “per capita cap” system for Medicaid: Instead of matching whatever states spend on Medicaid, the federal government would instead give them a fixed amount for each Medicaid enrollee.

Under the Senate GOP bill, that change would reduce federal spending about $772 billion over the next decade, according to the nonpartisan Congressional Budget Office. The House GOP version, which passed last month, would cut federal spending $880 billion over the same time period.

Republican proponents argue that controlling federal spending would force the healthcare system to become more efficient in providing services. A spokesman for Senate Majority Leader Mitch McConnell, R-Ky., referred a request for comment to the office of Sen. Orrin Hatch, R-Utah, chairman of the Finance Committee.

Katie Niederee, a spokesperson for the committee, said that the bill “reflects Republican priorities to bend the cost curve on federal entitlement programs and encourage states that tend to spend beyond their means to actually stay within their budget.”

Democrats believe that the nation’s neediest will be denied essential services — including in schools.

“No matter how they try to spin their massive cuts to Medicaid to make the Senate version look less ‘mean,’ it is clear that Trumpcare would mean massive cuts to schools and districts and massive pain for students and families,” said Sen. Patty Murray of Washington, the top Democrat on the Health, Education, Labor and Pensions Committee.

The Senate bill, known as the Better Care Reconciliation Act, exempts some of the most disabled children from the per-capita caps, but the number of children who would be affected is not known. Education advocates that whatever the exact number, schools will be in the same pinch.

“Health care will be rationed and schools will be forced to compete with other critical health care providers — hospitals, physicians, and clinics — that serve Medicaid-eligible children,” a coalition of more than 60 education, civil rights and child-welfare groups wrote to senators on Tuesday, urging them to reject the GOP legislation because of the impact on schools.

The Republican plan for Medicaid is likely to hurt schools in several ways, said Sasha Pudelski, who tracks healthcare policy for AASA, the superintendents’ association. Most directly, states may decide to prohibit schools from receiving Medicaid dollars because of what is likely to be stiff competition for limited resources, she said.

Less directly, states struggling to cover healthcare costs now covered by the federal government would have to seek cuts elsewhere in their budgets, including in education, which accounts for a large share of many states’ spending.

“The kids who will be hurt first and foremost are special ed kids and kids in poverty, but then everybody will be hurt, because we’ll have to shift dollars from the general education budget,” she said.

Schools receive less than 1 percent of federal Medicaid spending, according to the National Alliance for Medicaid in Schools. But federal Medicaid reimbursements constitute the third-largest federal funding stream to public schools, behind $15 billion they receive each year for educating poor children and $13 billion they receive to educate students with disabilities under the Individuals with Disabilities in Education Act (IDEA).

The federal government initially promised far more financial support for IDEA, the four-decade-old law that outlines schools’ obligations to educate students with disabilities. Congress pledged to pick up 40 percent of the cost of special-education services under the law, yet has never come close. It now pays only about 15 percent.

Medicaid payments have helped fill that gap. Without those dollars — and facing a recent Supreme Court decision that raised the bar for the services school districts owe students with disabilities — many districts wonder how they will pay for services they now provide.

In the small town of DuQoin, Ill., about 200 of 1,500 students receive special-education services. Medicaid helps pay the salaries of social workers, speech therapists and school nurses, as well as transportation for students with disabilities.

“If we’re not able to access the resources we have now … we’re going to be hurting kids,” said Superintendent Gary Kelly. DuQoin schools stand to lose about $170,000 in Medicaid reimbursements — a small but important fraction of the district’s $15 million annual budget, Kelly said.

Thanks to a political deadlock in the state capital, Illinois is headed into its third year without a budget, and schools across the state — including in DuQoin — are already scrimping, having cut teaching positions and other costs to balance their books. Illinois schools receive $286 million per year in Medicaid reimbursements, more than any other state except New Jersey and Texas, according to 2015 federal data.

In Crawfordsville, Ind., with a district of 2,400 students, Medicaid helps pay for a full-time registered nurse at each school. Those nurses provide primary care for many low-income students who don’t have a doctor of their own, Superintendent Scott Bowling said.

They also provide a first response to growing mental-health needs, and they care for children with complex medical needs, suctioning tracheostomies, tube-feeding children with gastric tubes, administering breathing treatments and emergency medication for seizures.

Bowling said that without the $50,000 in reimbursements, he’d have to lay off at least one nurse, and schools would be left without full-time nurse coverage. That gives him pause when he thinks about the emergencies his nurses have confronted: One saved a little boy from a severe asthma attack, administering an Epi-Pen to open his airways as they waited for medics to arrive. Another said this year she had intervened with two children who were suicidal, urging their parents to seek help immediately.

“Our nurses have literally saved students’ lives over the past few years — lives that we may have lost if we had to call them in from another site,” Bowling said.