WASHINGTON—Some people without health insurance will begin getting bills for Covid-19 treatments and testing after the Biden administration on Tuesday starts winding down a federal program that reimburses providers for virus-related care for the uninsured and that officials say is running out of funds.

The White House says it will end the reimbursement program, which started under the Trump administration and also pays hospitals and other healthcare providers for things such as administering Covid-19 vaccines to uninsured people, by the end of April because it is running out of money. The administration and hospitals are urging lawmakers to approve more funding for the program.

The White House earlier this month pushed for $22.5 billion in Covid-19 funding to pay for a range of programs, including the provider-relief funds and replenishing supplies of vaccines and antibody drugs. Congressional Republicans have called to repurpose Covid-19 money that states haven’t yet spent, but a number of House Democrats have balked at using money promised to states.

The provider relief fund, which was created to help hospitals and community health centers seeing decreased revenue or increased expenses because of the pandemic, was launched with more than $100 billion in 2020, and later legislation added about $78 billion. All of the money in the provider relief fund has been allocated, though it has not all been paid out, according to an official with the Department of Health and Human Services’ Health Resources and Services Administration.

About 50,000 hospitals, doctors and other providers have submitted claims requesting funding from the uninsured program since April 2020, the federal official said. The program bars providers who receive the funding from billing the uninsured for the difference between the money they get and the costs of treatment, a practice known as balance billing.

In total, about $20 billion from the relief fund has helped cover providers’ costs for administering vaccines, testing and treatment for people without health coverage.

The administration said it will stop accepting claims for treatment and testing for uninsured people Tuesday, and the deadline for claims for administering vaccines is in two weeks.

Though Covid-19 cases have declined sharply, funding is needed to prepare for future variants, the administration says.

Photo: Gabby Jones/Bloomberg News

After that, the medical bills for uninsured Covid-19 patients will depend on each hospital’s financial-aid policy and their prices, both of which can vary widely from one hospital to another. Financial aid is offered by most hospitals and is usually granted based on patient income, but hospitals largely choose their own cutoffs for eligibility. Some states set requirements. Prices for the same services are also sharply different across hospitals, with the uninsured often facing the highest prices.

The White House said it notified Congress in February that the fund that reimburses doctors and other medical providers for caring for uninsured individuals would stop taking new claims in March.

Hospitals that have relied on the funding say they are still feeling the financial squeeze of the pandemic and that more funding is needed for the uninsured program.

“We support the administration’s request for additional funding to ensure that the healthcare system has the resources it needs to continue to care for their patients, especially as we continue to manage Covid-19 in communities across the nation,” the American Hospital Association, a trade group for the industry, said. “Indeed, while we have made great progress against the virus, nearly 2,000 Americans passed away last week with Covid-19.”

An agreement between congressional Republicans and Democrats to repurpose unspent state Covid-19 funds as part of a smaller, $15.6 billion package was recently pulled from an omnibus spending bill after some Democrats balked at using the money originally intended for states.

Senior Senate Republicans say the administration hasn’t provided enough clarity on where funding that has already been allocated is going.

“Unless it’s paid for and it’s something that 10 Republicans will vote for, it’s hard to see how it passes the Senate,” Senate Minority Whip John Thune (R., S.D.) said recently.

Senior administration officials said Monday that they have briefed members of Congress on the status of Covid-19 relief funding since October and held more than three dozen meetings with lawmakers from both parties since January about the need for additional money. They said that $300 billion of the 2021 American Rescue Plan’s $1.9 trillion remains unspent but that $240 billion of it has been promised to cities and states that have built it into their budgets and that the remaining $60 billion is intended for emergency use.

Among health systems that benefited most from the aid are safety-net hospitals in states that didn’t expand Medicaid, such as Harris Health System in Houston, which received $196 million in relief funds for treatment of uninsured Covid-19 patients, according to government data published as of March 3.

The cuts could prompt Harris Health to curb other services “to absorb the cost for what is a federal responsibility,” Esmaeil Porsa, the system’s chief executive, said.

An estimated 9.6% of the population, or 31.1 million people, lacked health insurance in the first six months of 2021, according to the Centers for Disease Control and Prevention.

The Biden administration has said that while Covid-19 cases have sharply declined, funding is needed to prepare for future variants and continue with vaccination campaigns and other programs to protect the public.

The White House says the lack of new congressional funding means it won’t be able to purchase a second round of boosters for the general public, should federal regulators authorize another dose of the vaccine. Pfizer Inc. and partner BioNTech SE have asked the Food and Drug Administration to approve a fourth dose—or second booster shot—for people 65 years and older, while Moderna Inc., has sought approval for an additional booster for all U.S. adults.

Administration officials say they also expect the funding issue to impact the supply of monoclonal antibodies. They are also closely monitoring a new variant, BA.2, that has triggered an increase in cases overseas.

“Our concern right now is that we are going to run out of money to provide the types of vaccines, boosters, treatments to the immunocompromised and others free of charge that will help continue to battle” the pandemic, White House press secretary Jen Psaki said Monday.

—Melanie Evans contributed to this article.

Write to Stephanie Armour at stephanie.armour@wsj.com and Sabrina Siddiqui at Sabrina.Siddiqui@wsj.com