The nation's major banks cut their prime lending rate yesterday from 10 to 9 1/2 percent, the lowest level in almost seven years, and many financial analysts expect further reductions.

The cut in the prime rate, which started at Morgan Guaranty Trust Co. and quickly spread throughout the industry, touched off heavy selling on foreign exchange markets that sent the dollar lower, while stocks climbed rapidly on the news.

Many money market traders think the Federal Reserve will cut the discount rate soon, and if it does, they expect further reductions in the prime rate.

Meanwhile, the Commerce Department reported yesterday that the start of construction of new housing took its biggest drop in more than a year last month, slipping 14 percent to an annual rate of 1.66 million units from 1.93 million in April.

However, builders and housing economists predicted that the rapid decline of mortgage interest rates over the past six weeks will reverse the building slide, keeping both sales and construction activity at healthy rates at least through the summer.

The prime rate cut follows a general decline in short-term money market interest rates in recent weeks, a drop that has reduced the banking industry's cost of obtaining the money it lends.

Interest rates have been dropping, except for a brief period last winter, for about a year as a result of slowing economic growth and actions by the Federal Reserve to provide sufficient money to bolster the economy.

Short-term rates are between 3 and 4 percentage points lower than they were at this time last year, and about 1 1/2 percentage points lower than in March. Long-term rates, more critical for business investment, are down about 2 1/2 percentage points over the year but remain unusually high relative to current levels of inflation.

The drop in interest rates helped stock prices gain yesterday. The Dow Jones industrial average rose 6.38 points, closing at 1,304.77.

At the same time, the interest rate drop pushed the U.S. dollar lower on foreign exchange markets, with the West German mark gaining more than 1 percent in value. A controlled fall in the dollar's value is one goal of the Federal Reserve's current effort to move interest rates lower, according to Fed officials.

Money market rates are now low enough that they would be consistent with a prime rate lower than 9 1/2 percent, based on past spreads between the banks' cost of funds and the prime.

The market rates also are consistent with a 7 percent Federal Reserve discount rate -- the interest rate it charges on loans to financial institutions -- rather than the current 7 1/2 percent rate.

"There has been good customer demand" for Treasury securities at the new lower rates, said an analyst at a major New York government securities dealer. "The market is in good shape."

The prime lending rate, which used to be the rate banks charged their most creditworthy customers, has become more of a "reference rate" to which the rate charged on many business loans and a few consumer loans is tied.

However, most consumer loan rates are not directly affected by the prime. Nevertheless, the general drop in interest rates has caused some banks to begin to reduce their consumer loan rates, too.

As the Federal Reserve tightened its monetary policy in the first half of 1984 in the face of very rapid economic growth, the prime rate moved up from 11 percent to 13 percent.

After economic growth slowed abruptly in the third quarter of last year, the rate dropped steadily to reach a level of 10 1/2 percent at the end of January.

The last prime rate cut to 10 percent came on the heels of a reduction in the Fed's discount rate on May 17.

Sentiment among money market participants appeared to be divided yesterday, the New York analyst said, between those who thought market rates would continue to decline and those who thought they could rebound somewhat.

The Fed stepped in yesterday to reduce the availability of bank reserves when the federal funds rate -- the rate financial institutions charge one another when they lend reserves -- fell below 7 percent. Normally, the Fed seeks to keep the federal funds rate at least slightly higher than the discount rate.

However, the funds rate has been lower than the 7 1/2 percent discount rate for several days, one reason that analysts think a discount rate cut is coming.

Yesterday's action by the central bank appreared to indicate that while the Fed has sought to encourage lower short-term market interest rates generally, it does not want them to fall below current levels -- at least for now.

The Commerce Department will release May figures on personal income and consumption spending today and its so-called flash estimate of second-quarter gross national product on Thursday. Analysts were hoping the numbers would provide new clues to the course of the economy in coming months.

A common forecast for GNP growth for the current quarter is about a 3 percent annual rate, after adjustment for inflation. That would be higher than the first quarter's 0.7 percent rate, a number some forecasters think will be lower after a revision also set to be released Thursday.

Timothy Howard, chief economist of the Federal National Mortgage Association, said that "disappointing" housing sales in April caused builders to turn more cautious. He noted that the drop in sales followed a couple of months of rising mortgage rates, which leveled off at about 13 1/4 percent for fixed-rate 30-year mortgages in April. Now they "have finally come down" so that sales "ought to be a good deal healthier, and if that happens, starts ought to turn up," Howard said.