Treasury Secretary Donald T. Regan yesterday predicted that interest rates will stay much higher than assumed in the administration's budget numbers.

Responding to reports that Treasury Undersecretary Beryl Sprinkel said rates could stay fairly high at least into next year, Regan said he did not expect rates to go below the 15 percent range for some time, after their recent sharp run up to 20 percent.

In contrast, the administration has assumed for budget purposes that there will be an early fall in interest rates, which will cut the cost to the government of servicing its huge debt.

A spokesman for the Office of Management and Budget yesterday said there had been no re-estimate of the official interest rate forecast. But he added that if Sprinkel were right, this would obviously add to the interest expenses in the budget.

A White House spokesman said there were no comment on Sprinkel's remark, made in Gabon where the International Monetary Fund Interim Committee is meeting.

Meanwhile, rates climbed still higher yesterday, with several analysts predicting that the key prime lending rate would go up again today. Yesterday's rate movement came as the Federal Reserve apparently tightened credit further, bankers said.

Recent turmoil in the financial markets sparked off a search for extra spending cuts for this year, and encouraged the administration to move toward compromise on a cheaper tax bill, sources have said.

But so far no new cuts have been announced, and rates have continued to rise.

The president himself is believed to be concerned about soaring interest rates, although he has urged the Fed to keep a tight grip on the money supply. It is partly the Fed's attempt to rein in money growth, and hold it below the rate of inflation, that has led to present tight credit conditions and rising interest rates.

Earlier this week, Fed Chairman Paul Volcker met with the president and some top White House aides. Volcker had requested the meeting before the president was shot last March, according to administration officials. There has been speculation that the chairman did so in response to administration criticism of the Fed's implementation of money policy. Sources close to participants at the meeting indicated later that the meeting has gone badly.

Volcker, who has refused to comment on the meeting, has stressed the importance of reducing the budget deficit so that both fiscal and money policy are aimed at fighting inflation. The administration says that once Congress has enacted its budget cuts, interest rates will come down as financial markets revise their expectations of future inflation.

It has criticised the Fed for allowing fluctuations in money growth. Big rises in the money supply have led to higher inflation and higher interest rates, administration officials say.

The Fed has recently announced that it intends to exert even more short-term control over money growth. The policy-making Open Market Committee set a 5 1/2 percent annual rate of growth in the second quarter of the year for M-1B, a main measure of the money supply, at its meeting on March 31, the Fed reported yesterday.

It set a target range for the Federal Funds interest rate of 13 percent to 18 percent, down slightly from the 15 percent to 20 percent in the first quarter. However, after another meeting on Monday morning, market analysts believe that the Fed decided to tighten further. It did not supply funds to the market when the Fed funds rate topped 20 percent yesterday.

Chemical Bank and Marine Midland Bank each raised their broker loan rates one percentage point yesterday, to 20 1/2 percent and 20 percent, respectively. Changes in this rate often foreshadow changes in the prime.

Meanwhile, more evidence of an economic slowdown came yesterday as the Commerce Department reported that new factory orders for durable goods fell by 0.4 percent last month, after seasonal adjustment. New orders for nondefense capital goods dropped 5 percent in the month, to $22.6 billion after a March rise of 15 1/2 percent, the report said.

William C. Melton, economist at Irving Trust, a large New York bank, was reported as saying yesterday, "A 21 percent prime is almost a foregone conclusion, and we'll probably see it Friday."