WASHINGTON — Four federal appeals courts have upheld efforts by the Obama administration to guarantee access to free birth control for women, suggesting that the government may have found a way to circumvent religious organizations that refuse to provide coverage for some or all forms of contraception.

While pleased with the rulings, administration officials are not celebrating.

Another federal appeals court, based in Denver, is considering a challenge to the same federal policy by the Little Sisters of the Poor, an order of Roman Catholic nuns who operate homes for low-income older people. At a hearing in December, a three-judge panel asked tough questions of lawyers on both sides of that case.

Moreover, lawyers say, even if the birth control policy is ultimately upheld in court, the government could face practical problems lining up third parties to pay for contraceptives over an employer’s objections.

Sister Constance Veit, a spokeswoman for the Little Sisters of the Poor, said: “We live our lives very far from the public eye. We are not political activists at all. But we fear that the mandate could set a precedent for greater government intrusion into our ministry.

“Our health plans have never included contraceptives, and we’ve never had any complaints,” Veit said. “It has never been an issue for our staff.”

The appeals court decisions, in favor of the government, involve a wide range of nonprofit religious organizations that are eligible for a dispensation from rules requiring employers to cover birth control for women. On Friday, the administration offered similar relief to certain closely held for-profit corporations that oppose covering contraceptives because of the owners’ religious beliefs.

Some nonprofit religious groups are still battling the administration in court, and some for-profit employers might do so, but the appeals court decisions will bolster the government’s position.

In a typical opinion, Judge Richard A. Posner of the 7th US Circuit Court of Appeals in Chicago wrote this month, “When Wheaton College tells us that it is being ‘forced’ to allow ‘use’ of its health plans to cover emergency contraceptives, it is wrong.”

In fact, Posner wrote, the college — a Christian school in Illinois — is required only to inform its insurers or the government “that it will not use its health plans to cover emergency contraception, that it is out of the loop.”

Under the Affordable Care Act, employers with 50 or more full-time employees are generally required to offer health coverage or pay substantial penalties. Under federal rules, the insurance must cover preventive services, including all forms of contraception approved for women by the Food and Drug Administration.

Houses of worship, including churches, temples, and mosques, are automatically exempt from the contraceptive coverage requirement and do not have to file any paperwork.

Another group of employers — nonprofit religious organizations that object to such coverage on religious grounds — are eligible for an “accommodation”: They are deemed to comply with the requirement if they report their objections to their health insurance companies and the administrators of their health plans, or to the secretary of health and human services.

This arrangement, which some judges call an opt-out mechanism, is available to certain schools and universities, hospitals, and charities.

Many Roman Catholic, Baptist, and other religious groups have challenged the opt-out procedure, saying it makes them complicit in providing contraceptive coverage. But so far they have lost in federal appeals courts in the District of Columbia, Philadelphia, Chicago, and New Orleans.

“The plaintiffs have not shown a substantial burden on their religious exercise,” Judge Jerry E. Smith wrote last month in an opinion for a three-judge panel of the 5th US Circuit Court of Appeals, in New Orleans.

Plaintiffs in that case, led by East Texas Baptist University, argued that the opt-out procedure violated a federal law, the Religious Freedom Restoration Act of 1993, because it forced them to “facilitate” the coverage of contraceptives.

When an employer objects to providing contraceptive coverage for religious reasons, the insurer or administrator of its group health plan is supposed to pay for the contraceptives, at no cost to women or the employer. The insurer must make clear to women that their employer does not provide or pay for contraceptive benefits. The federal government has several ways to reimburse insurers.

After analyzing these requirements, the US Court of Appeals for the District of Columbia Circuit rejected challenges by Priests for Life, Catholic University of America, and other Roman Catholic groups.

“Religious nonprofits that opt out are excused from playing any role in the provision of contraceptive services, and they remain free to condemn contraception in the clearest terms,” wrote Judge Cornelia T.L. Pillard of the appeals court here. The Affordable Care Act, she wrote, “imposes an independent obligation” on insurers and third-party administrators to provide coverage.

Adam C. Jed, a Justice Department lawyer, explained that when a nonprofit religious group refuses to provide contraceptive coverage, “the government then steps in and fills the gap.”

But that arrangement is abhorrent to some religious groups like the Little Sisters of the Poor, because the government can rely on information provided by the employer.

“The action of the religious employer is the trigger for contraceptive coverage,” said Mark L. Rienzi, a lawyer at the Becket Fund for Religious Liberty, who represents the nuns. “The government wants to use our plan and our plan’s information to provide this coverage.”

Some judges disagree.

“Far from ‘triggering’ the provision of contraceptive coverage,” the opt-out is a way for employers to declare that they “will not be complicit” in providing such coverage, wrote Judge Marjorie O. Rendell of the 3rd US Circuit Court of Appeals in Philadelphia.

Critics of the contraceptive coverage requirement won a major victory when the Supreme Court ruled last year in favor of Hobby Lobby Stores and two other closely held family-owned corporations that objected to certain forms of birth control. Forcing those companies to provide contraceptive coverage to their employees violates the 1993 law protecting religious freedom, the court said.

Justices Samuel A. Alito Jr. and Anthony M. Kennedy praised the opt-out procedure for nonprofit groups as an alternative to the all-or-nothing mandate imposed on companies like Hobby Lobby.

The opt-out “achieves all of the government’s aims while providing greater respect for religious liberty,” Alito wrote.

Federal appeals courts have cited those comments as a reason to uphold the administration’s latest effort to guarantee access to contraceptive coverage for women insured by Roman Catholic hospitals, universities, and charities.