House-Senate conferees have agreed in the pending tax bill to improve the Social Security system's financial state by $15 billion over the next three years, but this may only buy Congress and President Reagan a few months to solve the ailing system's problems.

The breathing room for Social Security would come through cuts in Medicare, plus a requirement that federal employes start paying the Medicare part of the Social Security tax.

Instead of starting to run short of money at the start of 1984 as previously expected, the system might be able with these adjustments to keep going well into that year, according to experts at the House Aging Committee and elsewhere.

That would give Congress, the White House and a presidential commission studying the Social Security financing problem a little extra time to make decisions -- but only a little, experts say.

"It could buy you a number of months' extra time, perhaps not a year," said Laurie Fiori, legislative representative of the American Association of Retired Persons.

Social Security is by far the nation's largest social program. Its combined old-age, disability and Medicare payments to 36 million people, a seventh of the nation's population, will exceed $217 billion next year, more than a fifth of the federal budget.

The system's problems have become one of the nation's most inflamed political issues.

Last year, when Reagan proposed major benefits cuts, Democrats said he was seeking much more than needed to save the system and was trying to balance the budget on the backs of the aged.

Reagan withdrew his proposals and appointed the special commission, headed by economist Alan Greenspan; its recommendations will be made after the election.

Before the tax bill, it was estimated that the old-age fund, the largest and most in trouble, would start running short toward the end of this year but could keep going until the end of 1983 or into early 1984 by borrowing from the Medicare and disability funds.

The tax bill conferees agreed on included Medicare benefits cuts -- mostly new limits on payments to hospitals and doctors -- that are expected to hold program outlays $12.8 billion below their levels otherwise in fiscal years 1983 through 1985 and perhaps an equal amount over the following two years.

In addition, the bill would produce nearly $3 billion net new income to the Medicare trust fund over fiscal 1983 through 1985 and $2.5 billion more over fiscal 1986-87 by levying the Medicare tax on federal employes.

As a result, the old-age fund would be able to borrow more from Medicare and the whole system could keep going longer, perhaps past the middle of 1984, if the economy turns up better than expected.

Social Security's problems fall into short- and long-range. The combined old-age and disability trust funds are facing big difficulties right now, largely because the economy has performed much worse than expected since the 1977 Social Security amendments, with inflation outrunning the growth of wages.

Benefits are tied to inflation while receipts come from wages. The old-age fund is running out of money to cover all its obligations in a timely manner.

However, if the old-age and disability funds, as a unit, could get through the 1980s, they would start to improve rapidly beginning in 1990 as new taxes voted in 1977 kick in, and remain in good shape until about 2025 A.D.

Then, as a result of demographic changes, they would decline sharply as the large generation of post-World War II baby-boom children reached retirement age.

For Medicare the problem is entirely different. Medicare has a large surplus now and could help tide the old-age fund over in the next few years. But because hospital costs are inflating so sharply, the trust fund would run out of money rapidly and permanently, probably by 1987 or 1988.

For the old-age and disability funds, the special commission has been looking at such possible solutions as raising the normal retirement age to 68, cutting basic benefits levels for future retirees a bit below what they would be under current law, and holding future cost-of-living increases to less than current law allows.

For Medicare, a special advisory commission is about to be named; solutions proposed have included caps on reimbursements to hospitals and doctors and making benficiaries pay a larger share of the bills.