Chase Manhattan Bank lowered its prime lending rate to 15 3/4 percent yesterday, overtaking other major banks that dropped rates to 16 percent. Further cuts in the key prime rate, which is the base lending rate of major commercial banks for short-term loans to top corporate customers, are likely to follow soon, analysts said.

As the recession has worsened and choked off private-sector credit demand, short-term interest rates have fallen swiftly. Chase cited cheaper cost of funds and slackening loan demand in its decision to move to 15 3/4 percent.

Crocker National and Continental Illinois banks lowered their prime rates to 16 percent last week. Other major banks following them to that level yesterday included Citibank, Chemical Bank, and First National Bank of Chicago. Riggs National Bank of Washington also announced yesterday that it was lowering its prime to 16 percent.

As an indication of how much short-term rates have declined, the Federal funds rate averaged 12.94 percent in the week ending Nov. 11, down from almost 14 percent in the previous week and highs of about 20 percent in the summer. Yesterday the rate dropped to as low as 12 percent. This rate is what large banks charge each other for overnight loans, and is a key determinant of their cost of funds.

The prime has been falling more slowly than other rates, but is now down to its lowest level since November 1980. Since then, interest rates have been on a roller coaster, with the prime rising to a peak of 21 1/2 in December of last year, then falling to 17 percent in April before climbing back to 20 1/2 percent this summer. Willard C. Butcher, chairman of Chase Manhattan, cautioned this week against letting rates fall too fast this year. "If we now lower rates too fast and too far, we run a serious risk of a strong reaction and thus sharply higher interest rates."

The Federal Reserve Board does not want a repeat of last year when rates dropped very sharply leading a rapid increase in the money supply and quick rebound in the economy, which then sent interest rates climbing again. However, it also has been criticized for keeping money growth too tight this year.

Although senior Fed officials are pleased with the present decline in interest rates, they are concerned that if the economy picks up sharply next year then rates will rise again and perhaps stall the recovery. The administration predicts a steadily declining trend in rates.

However, many analysts fear that looming large federal deficits, coupled with the Fed's commitment to tight money, will send interest rates back up next year.