The Reagan administration is considering a plan to charge Medicare patients substantially more for their first 60 days of hospital care but to provide "catastrophic insurance" after their out-of-pocket costs reach a specified level, perhaps $2,500 or $3,000 a year, according to administration sources.

The "catastrophic" proposal would provide important new protection for seriously ill Medicare patients but, since relatively few of them would reach the specified level, costs would be more than offset by additional charges for the first 60 days.

According to one estimate, the package would save the government as much as $500 million in fiscal 1984 and as much as $1 billion in fiscal 1985.

The plan is being considered for inclusion in President Reagan's fiscal 1984 budget or a subsequent proposal on health care and is said to be close to final approval. The concept was outlined to the president about a week ago at a Cabinet meeting and he indicated that he likes the idea, sources said.

The plan has been discussed at various times in the last year, and its general outlines have been described to various officials on Capitol Hill and in the medical community.

Although no decisions have been made, sources said the proposal is increasingly gaining favor within the administration for three reasons:

* It will provide substantial budget savings at a time when the president has strongly indicated that he wants to continue cutting domestic programs.

* It will tie these cuts into a single package with a politically appealing "catastrophic" plan that will make beneficiaries feel they are gaining something, not merely being charged more money.

* It will improve the condition of the Medicare trust fund, which is expected to go bankrupt within the next decade or so because of rapidly rising hospital costs.

Government planners are looking at other possible Medicare cuts to shave the budget. These include slowing the growth of physician reimbursements and increasing the $75 that the Medicare patient must pay annually for doctor bills before the doctor-insurance section starts paying.

But the plan to increase the patients' share of hospital costs and provide catastrophic insurance is by far the most dramatic.

Medicare is the giant program that provides health benefits to about 29 million aged and disabled Social Security recipients. Its total costs exceed $55 billion a year, about one-sixth of the entire national health bill, public and private. It is the second biggest government benefit program, behind its big brother, Social Security, and it is one of the fastest-growing programs in the entire government.

Under the existing program, a Medicare patient entering a hospital pays the first $260 of the hospital bill, basically equivalent to the first day's fee. Then the government pays the entire cost of the second through 60th days. After the 60th day the patient pays a quarter of the daily cost.

Basically, the plan under study would require Medicare patients to pay the first-day charge -- which will rise automatically to $304 Jan. 1 -- plus between 6 and 10 percent of the cost of each additional day. This would save the government hundreds of millions of dollars a year.

Under the "catastrophic" proposal, once a patient's total out-of-pocket outlays for all covered services under Medicare reach a specified figure in the range of $2,500 to $3,000 a year, the patient would have no further out-of-pocket expenses for any Medicare service.

The government's saving could easily be set higher or lower by raising or lowering the amount the patient pays after the first day and by altering the "catastrophic" ceiling.