Faith Rojas works on a puzzle with Matthew Kidder, who has cerebral palsy, at Chimes Virginia in December. Chimes provides community services for children and adults with intellectual and developmental disabilities. (Evelyn Hockstein for The Washington Post)

A Virginia effort to bring ­government-funded services to thousands more disabled people could, paradoxically, leave those in the state’s largest jurisdiction without funding for some types of aid.

Fairfax County officials say they are likely to create a new waiting list or restrict aid for day-support and work programs because of a redesign of the state’s Medicaid waiver program that — in an effort to comply with the Americans With Disabilities Act — requires the county to accept several thousand people who weren’t previously eligible for government aid.

The county provides about $24 million for day-support and work programs for about 4,300 people who have not yet received Medicaid vouchers, or waivers, which cover the cost of care. Until recently, the only people eligible for such county funding were intellectually disabled people with IQ scores of 70 or below.

The Medicaid waiver redesign — which creates one funding stream for all disabilities — means the county must also cover disabled people with higher IQs or not offer to fund services for anyone, Fairfax officials say.

The county estimates that the broader pool of people eligible for services could increase the cost of covering those who don’t have vouchers to at least $40 million per year by 2021 — a heavy burden in an era of budget constraints that have made it harder to find money for schools, police and other priorities.

“There is going to need to be a lot of work done with those families to look for ways in which we can accommodate a larger population within our budget,” said Fairfax County Board of Supervisors Chairman Sharon Bulova (D). “It will probably entail addressing service we provide based on a level of need.”

The problem is part of a broader challenge in Virginia of providing services to a disabled population that continues to grow, as diagnoses increase for autism and other ailments and people with all types of disabilities are living longer lives.

Nearly 12,600 people in the state receive Medicaid reimbursements for disability-related services.

An additional 11,525 disabled people are on a state waiting list for Medicaid waivers. That is an increase of about 15 percent since July 2015, despite efforts in Richmond to reduce the wait list as part of a 2012 federal court settlement that requires Virginia to offer people with disabilities opportunities to live and work in communities instead of institutional settings.

The redesigned Medicaid waivers are meant to bring more people into the system by covering a wider array of services than the old waivers and prioritizing aid based on need, state officials say. Previously, the first-come, first-served system included separate funding streams for people with different categories of disabilities.

In Northern Virginia, the cost of care is generally higher than in the rest of the state and can climb into the tens of thousands of dollars in annual costs per person.

Although Arlington County and Alexandria also provide local funds for day-support and work programs, Fairfax is most affected by the waiver redesign because it is the state’s largest jurisdiction and home to the largest number of disabled residents in the state.

The county has set aside $1.6 million in reserve funds to address next year’s expected increase in demand. Officials say they may need an additional $5.1 million.

Another option, they say, is to create a second waiting list for those who are already waiting for Medicaid vouchers and would also have to wait their turn for county funds.

Advocates for the disabled argue that Fairfax instead should gear its local aid toward less expensive types of services, which could cover more people.

Lucy Beadnell, director of advocacy for the Arc of Northern Virginia, said her group is pushing for the county to pay more toward work programs, which on average cost between $7,400 and $18,500 per year per person. Annual costs for day-support programs, which involve care in group settings, are about $27,000 per person, according to county figures.

“That is a huge funding difference,” Beadnell said. “Why aren’t we pushing more people toward employment, and making sure that they understand that anybody can be successful with the right support, instead of continuing this model of warehousing people?”

Jean Hartman, assistant deputy director of the Fairfax-Falls Church Community Services Board, which administers aid to the disabled, said her organization agrees that more people should be enrolled in work programs.

But not enough of those receiving aid function at a high enough level, she said.

“We have people with really intense medical needs who are not able to communicate all that well,” Hartman said. “The challenge is: How do we manage that?”