The Trump administration has labored zealously to cut federal regulations, but its latest move has still astonished some experts on health care: It has asked for recommendations to relax rules that prohibit kickbacks and other payments intended to influence care for people on Medicare or Medicaid.

The goal is to open pathways for doctors and hospitals to work together to improve care and save money. The challenge will be to accomplish that without also increasing the risk of fraud.

With its request for advice, the administration has touched off a lobbying frenzy. Health care providers of all types are urging officials to waive or roll back the requirements of federal fraud and abuse laws so they can join forces and coordinate care, sharing cost reductions and profits in ways that would not otherwise be allowed.

From hundreds of letters sent to the government by health care executives and lobbyists in the last few weeks, some themes emerge: Federal laws prevent insurers from rewarding Medicare patients who lose weight or take medicines as prescribed. And they create legal risks for any arrangement in which a hospital pays a bonus to doctors for cutting costs or achieving clinical goals.

The existing rules are aimed at preventing improper influence over choices of doctors, hospitals and prescription drugs for Medicare and Medicaid beneficiaries. The two programs cover more than 100 million Americans and account for more than one-third of all health spending, so even small changes in law enforcement priorities can have big implications.

Federal health officials are reviewing the proposals for what they call a “regulatory sprint to coordinated care” even as the Justice Department and other law enforcement agencies crack down on health care fraud, continually exposing schemes to bilk government health programs.

“The administration is inviting companies in the health care industry to write a ‘get out of jail free card’ for themselves, which they can use if they are investigated or prosecuted,” said James J. Pepper, a lawyer outside Philadelphia who has represented many whistle-blowers in the industry.

Federal laws make it a crime to offer or pay any “remuneration” in return for the referral of Medicare or Medicaid patients, and they limit doctors’ ability to refer patients to medical businesses in which the doctors have a financial interest, a practice known as self-referral.

These laws “impose undue burdens on physicians and serve as obstacles to coordinated care,” said Dr. James L. Madara, the chief executive of the American Medical Association. The laws, he said, were enacted decades ago “in a fee-for-service world that paid for services on a piecemeal basis.”

Melinda R. Hatton, senior vice president and general counsel of the American Hospital Association, said the laws stifle “many innocuous or beneficial arrangements” that could provide patients with better care at lower cost.

Hospitals often say they want to reward doctors who meet certain goals for improving the health of patients, reducing the length of hospital stays and preventing readmissions. But federal courts have held that the anti-kickback statute can be violated if even one purpose of the remuneration is to induce referrals or generate business for the hospital.

The premise of the kickback and self-referral laws is that health care providers should make medical decisions based on the needs of patients, not on the financial interests of doctors or other providers.

Health care providers can be fined if they offer financial incentives to Medicare or Medicaid patients to use their services or products. Drug companies have been found to violate the law when they give kickbacks to pharmacies in return for recommending their drugs to patients. Hospitals can also be fined if they make payments to a doctor “as an inducement to reduce or limit services” provided to a Medicare or Medicaid beneficiary.

Doctors, hospitals and drug companies are urging the Trump administration to provide broad legal protection — a “safe harbor” — for arrangements that promote coordinated, “value-based care.” In soliciting advice, the Trump administration said it wanted to hear about the possible need for “a new exception to the physician self-referral law” and “exceptions to the definition of remuneration.”

Almost every week the Justice Department files another case against health care providers. Many of the cases were brought to the government’s attention by people who say they saw the bad behavior while working in the industry.

“Good providers can work within the existing rules,” said Joel M. Androphy, a Houston lawyer who has handled many health care fraud cases. “The only people I ever hear complaining are people who got caught cheating or are trying to take advantage of the system. It would be disgraceful to change the rules to appease the violators.”

But the laws are complex, and the stakes are high. A health care provider who violates the anti-kickback or self-referral law may face business-crippling fines under the False Claims Act and can be excluded from Medicare and Medicaid, a penalty tantamount to a professional death sentence for some providers.

Federal law generally prevents insurers and health care providers from offering free or discounted goods and services to Medicare and Medicaid patients if the gifts are likely to influence a patient’s choice of a particular provider. Hospital executives say the law creates potential problems when they want to offer social services, free meals, transportation vouchers or housing assistance to patients in the community.

Likewise, drug companies say they want to provide financial assistance to Medicare patients who cannot afford their share of the bill for expensive medicines.

AstraZeneca, the drug company, said that older Americans with drug coverage under Part D of Medicare “often face prohibitively high cost-sharing amounts for their medicines,” but that drug manufacturers cannot help them pay these costs. For this reason, it said, the government should provide legal protection for arrangements that link the cost of a drug to its value for patients.

Even as health care providers complain about the broad reach of the anti-kickback statute, the Justice Department is aggressively pursuing violations.

A Texas hospital administrator was convicted in October for his role in submitting false claims to Medicare for the treatment of people with severe mental illness. Evidence at the trial showed that he and others had paid kickbacks to “patient recruiters” who sent Medicare patients to the hospital.

The owner of a Florida pharmacy pleaded guilty last month for his role in a scheme to pay kickbacks to Medicare beneficiaries in exchange for their promise to fill prescriptions at his pharmacy.

The Justice Department in April accused Insys Therapeutics of paying kickbacks to induce doctors to prescribe its powerful opioid painkiller for their patients. The company said in August that it had reached an agreement in principle to settle the case by paying the government $150 million.

The line between patient assistance and marketing tactics is sometimes vague.

This month, the inspector general of the Department of Health and Human Services refused to approve a proposal by a drug company to give hospitals free vials of an expensive drug to treat a disorder that causes seizures in young children. The inspector general said this arrangement could encourage doctors to continue prescribing the drug for patients outside the hospital, driving up costs for consumers, Medicare, Medicaid and commercial insurance.