Hints of slower economic growth?

The pronounced slowdown in US economic growth, which many forecasters long have predicted, finally may be about to occur, new government data hint. The government's index of leading economic indicators, which is designed to predict future trends in the economy, moved down 0.9 percent in June, its first significant decline in two years.

Several months of declines are needed to confirm a trend, economists caution.

''The numbers bounce around a lot from month to month,'' says David Berson, senior economist at Wharton Econometric Forecasting Associates. ''Take any one month's numbers with a large grain of salt.''

Moreover, the index is subject to revision.

But the decline in the index is in line with many forecasters' expectations that the rate of economic growth will slow significantly in the next several quarters. And some economist say that any revision in June's index won't cancel out the decline.

Forecasters say a sharp slowdown in the economy's growth rate is needed to prevent inflation from rekindling. Slower growth also would mean less upward pressure on interest rates but also slower progress in reducing unemployment.

The drop in the index ''suggests a slowing of the economy,'' says Bernard Markstein III, senior economist at Chase Econometrics.

''A change in the economic weather is close at hand,'' says Wes English, based in San Ramon, Calif., and publisher of a real estate advisory letter. ''So far the barometers only indicate a dramatic slowdown is in the offing.''

The slowdown now apparently on the horizon, and recent declines in commodity prices, will not be enough to keep inflation and interest rates from picking up a bit during the rest of 1984 and into 1985, forecasters say. For one thing, more of the economy's productive capacity will be in use, pushing up prices. And business borrowing is expected to be brisk, putting upward pressure on interest rates.

However, a steady stream of negative signals from the index of leading economic indicators is not anticipated. generally accepted signal that a recession is approaching.

''At this stage in the recovery it is not unusual to have one month of falling (indicators) followed by rises,'' Mr. Markstein says.

Economic growth in the second half of 1984 will be slower than in the first half, explains Robert Gough, senior vice-president of Data Resources Inc., a forecasting firm. ''Consequently you can expect some declines in the (index) to reflect a slowing of momentum'' in the economy but not a steady pattern of decline that would signal a recession.

As if to emphasize the unpredictability of economic statistics, the government announced Tuesday that after seasonal adjustment, sales of new single-family homes rose at a 0.6 percent annual rate in June. The increase came after three straight months of declines. The rise was unexpected because mortgage interest rates have been rising. Fixed-rate mortgages now are being offered with interest rates near 15 percent.

The good news on sales was tempered by the fact that the supply of unsold homes grew to 341,000, a six-month supply and the highest the inventory of unsold new homes has been since September 1982.

The home-sale figure ''is a very weak number,'' says Michael Sumichrast, chief economist of the National Association of Home Builders. ''I have been cautioning builders not to build.''

A slowdown in housing is one reason forceasters say the economy's growth will slow. Last week the government said the economy had grown at a torrid 10.1 percent seasonally adjusted annual rate in the first three months of 1984 and estimated growth in the second quarter at a slower but still robust 7.5 percent.

For 1985 as a whole, Chase Econometrics expects the economy to grow 3.2 percent vs. 7.4 percent in 1984. Inflation, as measured by the consumer price index, is seen increasing 5.1 percent vs. 4.4 percent in 1984.

In the long run, the economy cannot grow faster than 3 to 3.5 percent without triggering higher inflation rates, economists say, because on average that is how fast the economy's productive capacity is growing. Higher prices are being forecast even though some commodity prices - like those for oil, gold, and certain farm products - have fallen. Forecasters say much of the decline in commodity prices is due to the dollar's unusual strength, which is not expected to continue. Labor costs, which play a large role in determining the inflation rate, also are expected to climb in 1985, partly as a result of major labor negotiations.

Increases in the amount a worker can produce in an hour are expected to slow next year, forecasters say. Nonfarm productivity grew at a 3.3 percent annual rate in the second quarter of the year, vs. 2.9 percent in the first quarter, the Labor Department reports.