President Reagan has proposed some of his deepest spending cuts next year in one of the fastest growing and most sensitive parts of the federal budget, the basic health care programs that now account for $1 out of every $10 the government spends.

The programs are Medicare for the elderly and disabled and Medicaid for the poor. Together they help pay the medical bills of one American in five, and, even with the $5.2 billion in cuts that Reagan wants to make, their net cost next year will be $68 billion, an increase of about $4.4 billion, or 7 percent, the government estimates.

The problem is that is less than the likely rate of inflation in the health-care sector. The Reagan proposals would thus mean either real cuts in services or shifts in cost to state and local governments, private insurers, doctors and hospitals and patients.

The proposals thus are sure to meet resistance in Congress, which already voted some Medicare and Medicaid cuts reluctantly last summer.

The nation's governors and mayors say they will fight many of the new proposals. So will the interest groups affected, ranging from the Children's Defense Fund and American Association of Retired Persons to the Federation of American Hospitals, which represents the nation's for-profit hospitals and will oppose cuts in reimbursement rates.

The administration says, however, that the president's proposals are necessary not just to reduce next year's deficit but as a first step in restructuring the health industry to make it more cost-conscious and less inflationary. It rejects the suggestion that the nation's health will suffer.

Here, in this important and contentious part of the budget, are the major changes proposed by Reagan: Medicare

* Two Percent Cut. The administration wants to reimburse hospitals 2 percent less for Medicare patients than under existing rules. The hospitals would be forbidden to recoup by charging Medicare patients supplementary fees. Saving: $653 million.

* Federal Employes. The Medicare portion (1.3 percentage points) of the Social Security tax would be imposed on federal employes who then would all be eligible for Medicare on the same basis as everyone else.

The justification: at present, about 80 percent of federal retirees qualify for Medicare at age 65 on the basis of their own or their spouse's past work in private employment. But many of them do so on the cheap, on the basis of short periods of private employment in which they pay relatively small amounts of Social Security tax.

Making them pay the tax throughout their working lives would make them pay their fair share for eligibility. Net increased federal income: $619 million.

* Hospital-Stay Targets. Blue Cross and health insurance companies would be asked to work with hospitals to set targets for reducing patient lengths-of-stay in hospital. The average Medicare stay is 11 days. Cutting this by only one-third of a day would save $330 million.

* Working Aged. Many employers cut health benefits for people who continue working past 65. The administration would require employers to offer workers age 65 to 69 the same health benefits as younger workers, and would make Medicare the secondary payer. Saving: $303 million.

* Fee Screens. Physician fee schedules are updated each July. Reagan would delay this until October and cut this year's increase from 8 percent to 5 percent. Savings: $245 million.

* Radiologists and Pathologists. The proposals would cut payments to these doctors for in-hospital patients from 100 percent of reasonable charges to 80 percent, and let them seek the rest from the patient. Saving: $160 million.

* Other Changes: Cut reimbursements for kidney dialysis ($130 million); automatically increase each year, by the same percentage the consumer price index goes up, the current $75 deductible that the patient must pay annually under the Part B doctor-insurance program before Medicare payments begin ($65 million). Medicaid

* Matching Rates. The federal government now reimburses the states for between 50 percent and 77.5 percent of their outlays for Medicaid (poor states get the higher amounts).

The administration wants to cut each state's reimbursement rate by 3 percentage points on that part of the state's outlays that go for optional services (not required under federal law) and for all "medically needy" clients, people who are not quite poor enough to qualify for mandatory Medicaid coverage but who the states opt to cover on grounds they are too poor to pay their medical bills. Two-fifths of the entire program cost is optional, for such services as intermediate nursing home care, podiatry and dental care. Saving: $600 million.

* Co-Payments. At present, states are barred from charging mandatory Medicaid patients (those on Aid to Families with Dependent Children or the Supplementary Security Income program) any fees for basic mandatory services.

The administration would require the states to charge both mandatory and medically needy recipients between $1 and $1.50 for each physician's visit and $1 to $2 a day for each in-patient hospital day. This provision "is designed to make patients more cost-conscious in their use of health-care services." Saving: $329 million.

* Family Responsibility. The administration would allow states to charge the children of Medicaid patients in nursing homes for part of the cost. Saving: $29 million. It would also allow states to seek liens on homes or other property of such patients while the individual is still alive, but they couldn't take it over as long as it was still needed as a residence by the recipient, or spouse or minor child. Saving: $183 million.

* Other: Cease paying states for any costs resulting from error rates exceeding 3 percent in 1983, 2 percent in 1984, 1 percent in 1985 and zero percent in 1986. Saving: $59 million.