The government's index of leading indicators, often a guide to future trends in the economy, dropped by 1.3 percent in June following a 1.5 percent decline in May, the Commerce Department reported yesterday.

Meanwhile, Murray Weidenbaum, chairman of the Council of Economic Advisers, told the Senate banking committee, "We are now experiencing a pronounced slowing" of key sectors of the economy.

Recent preliminary figures on the Gross National Product show the nation's total output slipping by 1.9 percent, annual rate, in the second quarter of the year.

The Reagan administration predicts, however, that the economy will pick up toward the end of the year. Weidenbaum said the tax bill now going through Congress will be a "powerful" impetus to faster growth. He also held hope of an imminent decline in interest rates.

Prolonged high interest rates have dealt a blow to the housing and auto sectors. More than half of the decline in the leading indicators in June was due to a slump in building permits, Commerce Department analysts said.

Weidenbaum was questioned by the banking committee on why the financial markets had not yet responded to the administration's economic program by lowering interest rates. He said markets "don't yet believe that Congress, the president and the Federal Reserve are going to stay the course" but added, "There is little doubt in my mind that we should begin to see, in the near future, a substantial unwinding of the inflationary premium that has been built into both short and long term interest rates over the past several years."

Weidenbaum repeated his claim that recent signs of a slowing in inflation were due primarily to fiscal and money restraint. Some financiers fear that a large tax cut will jeopardize the administration's goal of reaching a balanced budget by 1984.

Weidenbaum said it is up to individual firms and unions to agree on wage settlements without "jawboning" on the part of the administration. But they should recognize that "since inflation is moderating, current wage negotiation should take that into account."

If they make inflationary wage agreements, he warned, "firms . . . would see their profitability erode. And they should not look to this administration for help in such a situation."

On the tax bill, which now has been loaded with "sweeteners," Weidenbaum said four-fifths of its cost in 1986 would be for the president's two original proposals -- individual rate cuts and accelerated depreciation for businesses. Of the remaining fifth of the cost, much would be lost through changing the so-called marriage penalty and through the indexing provisions now included in the bill.

Economist David Cross, from Chase Econometrics, said the leading indicator figures were in line with other recent data on the economy, and with the forecasts of a downturn made by many analysts. "We are in an economic downturn" that will continue at least until the fourth quarter of the year, he said.

Cross predicted only a gradual recovery, contingent on falling interest rates.

The Commerce report showed that five of the available 10 indicators were down during June. After the building permits, lower prices for sensitive raw materials accounted for a further third of the overall decline.