Continuing high inflation, strong demands for credit by businesses, consumers and government, and an increasingly restrictive monetary policy will push interest rates to record hights next year, according to Salomon Bros., a New York investment banking firm.

The economy will continue to grow "well into 1979," but inflation and high interest rates will trigger a recession by the end of the year, according to the firm's chief economist, Henry Kaufman. Kaufman said he did not know whether the recession would be mild or severe.

He said the economy should grow by 2.5 percent or more next year, but inflation will remain very strong, even higher than it was this year. He predicted the inflation rate will end up between 9 and 101/2 percent next year despite the Carter administration's anti-inflation program.

Kaufman's growth predictions are close to those of both administration officials and Federal Reserve Board Chairman G. William Miller, although neither the administration nor Miller foresees a recession next year.

At a luncheon meeting with reporters today, Kaufman said it is difficult to predict just how high interest rates will climb next year, but said the prime rate "will reach and exceed 13 percent." The highest level the prime rate has reached so far was during the severe credit crunch of 1974 when it rose to 12 percent and stayed there for several months.

Today the prime rate, the interest banks charge their best corporate customers for short-term loans, stands at 111/2 percent.

The Salomon Bros. economist said that despite the near-record interest rates prevailing today, Federal Reserve Board monetary policy has not been very effective in restraining demands for credit and dampening the growth of the money supply.

Since late April, in an attempt to fight inflation and prop up a sagging dollar, the Federal Reserve has been trying to raise interest rates. The federal funds rate-the interest banks charge each other for overnight loans of excess reserves and the key Fed tool in its monetary policy operations-has risen from about 6 percent to 10 percent.

Even so, the Fed has not shown resolve in continuing to tighten monetary policy, Kaufman said. However, he predicted that a reluctant Federal Reserve eventually will push interest rates high enough to reduce credit demands. When that happens, some sectors of the economy will be squeezed out of the credit markets and a recession will ensue.

The annual Salomon Bros. report on financial prospects for next year said that credit demand by businesses, consumers and the government will reach record levels in 1979. Kaufman predicted that new credit demand will total $389 billion, up from $380 billion in 1978. Much of this increase in borrowing needs will be inflation-related, as the cost of buying and financing everything rises sharply.

In addition to inflation, however, the Salomon Bros. report predicted that continuing strong predicted that continuing strong economic growth during the first half of 1979 also will push up credit demamd.

Kaufman said that business spending on new plants and equipment should rise by 31/2 to 4 percent after adjustment for inflation. That is a much stronger increase in real capital spending than businessmen themselves predicted to the Commerce Department in a survey the department release last week.

At the same time and in part because of continued high employment through the first half of 1979, household demand for both mortgage and installment credit will remain strong as well during 1979.

Salomon Bros. estimated that despite President Carter's recent calls for spending restraint, the federal government will need to borrow $75 billion in 1979, more than the $72 billion it is expected to borrow this year.

Kaufman noted that the president's vow to hold down federal spending does not take effect until the start of fiscal 1980, which begins next October. For the nex 10 months, "Fiscal restraint is really lacking," he said.