The nation's economy appears on the verge of a slowdown following a report yesterday that the government's index of leading indicators fell in December for the first time in six months.

The Commerce Department reported that the index, which is supposed to foreshadow changes in the economy, fell 0.8 percent last month. This was the first decline since May, when last spring's recession was in full swing.

President Reagan said yesterday the news "underscores the need to turn this economy round" and to "deal simultaneously with inflation and a sluggish economy." He promised that his comprehensive economic package of tax and spending cuts, to be unveiled around Feb. 17, "is designed to do just that."

Citibank, the nation's second-largest bank, yesterday announced a drop in its prime lending rate from 20 percent to 19 1/2 percent. Shortly afterward the Federal Reserve published figures showing further sharp declines in money growth in the week ended Jan. 21.

Earlier this week Marine Midland Bank lowered its prime to 19 1/2 percent, but so far Citibank is the only other major bank to follow suit. A gradual decline in interest rates seems to have set in, as loan demand slows with a weakening economy.

This is reflected by the declines in money growth, although significant changes in the banking system, with the introduction of nationwide interest-bearing checking accounts, has made the money figures volatile and hard to interpret this month.

The narrow measure of the money supply, M1-A, dropped by a seasonally adjusted $3.2 billion in the latest week to $374 billion. M1-B, which includes all checking accounts and cash, dropped slightly to $416.4 billion. In the latest four weeks M1-A averaged an 8.8 percent annual rate of decline, while M1-B averaged a rise of 4.1 percent. [Tables on F3]

Another government report yesterday showed that productivity in American business dropped by 1.9 percent in the fourth quarter of last year, after a smaller rise in the previous months. For 1980 as a whole, productivity in private business ropped by 0.3 percent, the third consecutive year of decline.

Economists mostly agree that the 5 percent growth rate notched up in the last quarter of 1980 will be followed by a much flatter performance this quarter. Record-high interest rates reached in December are expected to slow the economy, particularly the housing and auto markets. But so far there have been few signs of the expected recession.

The drop in the leading indicators is the strongest suggestion so far that a widespread slowdown is underway. However Commerce Department economists warn that one month's figures should not be taken by themselves as firm indication of a trend.

Although the leading indicators turned down last month, the Commerce Department yesterday revised upwards the figures for November. These now show a 1.5 percent increase from the previous month, compared with an original estimate of a 0.9 percent rise.

Last month seven out of the 10 indicators which comprise the index contributed to its decline. These were: new orders after adjusting for inflation; contracts and order for plant and equipment, also after inflation adjusting; a drop in the money supply in real terms; building permits; stock prices; changes in some raw material prices which are sensitive to the level of demand in the economy; and changes in total liquid assets.

The productivity decline at the end of last year came as the number of hours worked in private business rose by 8.3 percent, while output was up by only 6.3 percent. Flat or sluggish productivity growth means there is little or no room for workers to have real increases in their standard of living.

In another report yesterday, the Agriculture Department said that farm prices dropped by 0.8 percent in January but were still 11 percent higher than a year ago.