Several of the nation's largest banks raised their prime interest rate to 9.75 percent yesterday amid signs that the Federal Reserve Board is taking further steps to raise short-term interest rates to fight inflation.

First National Bank of Chicago led yesterday's increases in the prime rate - the interest banks charge their best corporate customers for a short-term loan. It was the eight time the prime rate has been raised this year.

The increase, which most of the rest of the industry can be expected to emulate during the next few days, had been anticipated.

At the same time, money market analysts said, the Federal Reserve appears to have tightened monetary policy another notch by permitting the so-called federal funds rate of rise from 8.5 percent to 8 5/8 percent.

The federal funds rate is the interest banks charge each other for overnight loans of excess reserves. It is the key rate the central bank uses in conducting its open market policy by buying and selling government securities.

Only last week, the Fed increased the federal funds rate from 8 3/8 percent to 8.5 percent. Last Friday, the central bank boosted the discount rate to a record 8 percent from 7.75 percent. The discount rate is the interest the Fed charge member banks that borrow from it.

Since late April, the Fed has raised the federal funds rate - and indirectly all other short-term interest rates - by nearly two full percentage points. The rate averaged 6.75 percent in late April.

Despite the Fed's apparent tightening, however, money supply growth has continued to grow, outpace the rise in interest rates, and both business and consumer demand for credit remain strong. Analysts said that portends further increases in interest rates, including more rises in the federal funds rate and perhaps another boost in the discount rate soon.

Although the discount rate is no longer an important tool of monetary policy, the Fed uses it to signal its intentions.

The Fed raises interest rates to make borrowing more expensive and, in turn, reduce the rate of growth of the money supply. Most economists think that the rate of growth of the money supply - currency in circulation and checking accounts - is a key determinant of inflation.

Recently the Fed has been raising the federal funds rate in small, one-eighth percentage point steps and some analysts, such as Leonard Santow of J. Henry Schroder Bank and Trust Co., thinks the Fed has not gotten the impact it wanted from these small increases.

As a result, Santow said he thinks the Fed will push the funds rate as high as 8.75 percent soon.

As other market interest rates rise, the so-called prime rate has followed right along. The prime rate was at 7.75 percenta on Jan. 1, when the federal funds rate was about 6.5 percent.

The Fed has been pushing up interest rates not only to fight inflation, but to help shore up the international value of the dollar by making it attractive for Americans to borrow dollars abroad, soaking up some of the excess overseas dollars, and worthwhile for foreigners to invest their dollars in U.S. securities.