The chairman of the nation's fourth biggest bank said this week that he does not think the prime lending rate will rise, as predicted by the Reagan administration, even though the cost of funds to banks has climbed sharply in recent weeks.

In fact, John F. McGillicuddy, chairman of Manufacturers Hanover Trust Co., said he believes the prime rate, instead of rising from its current 10.5 percent, should be below 10 percent by the end of the year. Last week, Martin S. Feldstein, chairman of the president's Council of Economic Advisers, said he expected banks would increase their prime rates soon.

The prime rate is the rate banks use as a basis to calculate the borrowing charge to most of the business customers.

McGillicuddy told reporters at a luncheon meeting that if short-term rates continue to escalate, he might have to revise his prediction that banks will not boost the prime. However, he said, he thinks the sharp rise in so-called market rates in recent weeks is a "blip" and that the rates will subside shortly.

Big banks buy a large proportion of their deposits in the open market, where rates are set by depositors on a minute-to-minute bid and ask basis over a vast telephone linkup, in much the same way stock prices are determined in an ongoing auction on stock exchanges.

The prime rate, on the other hand, is an administered rate, set by banks themselves. Banks usually try to relate their interest charge to the price they must pay for deposits in the open market.

In recent weeks, the cost of three-month deposits has risen by a percentage point or more. Part of the increase in rates apparently is due to tighter monetary policy by the Federal Reserve, the nation's central bank, while some of the increase is due to investor fears that the central bank will tighten further.

For example, banks today must pay about 9.20 percent for three-month deposits and more than 10 percent for six-month deposits. McGillicuddy admitted that "spreads" (the difference between what a bank pays for its funds and what it charges borrowers) have "narrowed appreciably."

McGillicuddy said that because bank profits have been strong the last six months, banks are under less pressure to boost the prime lending rate than they would be if earnings were weaker. He said that banks are also aware that interest rates play a key role in the economy and that a boost in bank charges might take "some steam" out of the recovery.