WASHINGTON — 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.