The economy will grow between 3 percent and 4 percent in real terms in the final three months of the year, with interest rates declining in a sawtooth pattern, Treasury Secretary Donald T. Regan predicted yesterday.

Regan also told reporters he hoped unemployment would peak this month, but he cautioned that it was too soon to tell whether the September rate -- to be published Oct. 8 -- will prove to be the peak. Many experts fear that the jobless rate will breach the 10 percent mark in the September numbers.

Despite the belief of several analysts that the economy has not yet begun to recover, Regan said at the breakfast meeting yesterday, "I think we're getting a slow recovery here" that is under way principally in the service sectors of the economy, such as the hotel and restaurant business, and in financial institutions. He said there is no doubt that in the short run, there is a trade-off between inflation and unemployment, although he denied that the administraton had deliberately pushed the jobless rate up to curb inflation.

The Treasury secretary said he would like to see economic growth of 4 1/2 percent next year, although this was not a forecast. The administration's last official forecast was for a 4.4 percent rise in real output next year, but Regan last week admitted that this now looked too high. Private forecasters on average predict growth between 2.9 percent and 3.9 percent next year. Regan commented yesterday that a 4 to 4 1/2 percent rate of increase in real Gross National Product would be "mild" compared with previous post-war recoveries.

The budget deficit for fiscal 1983, which begins Friday, will probably be larger than the 1982 deficit, now expected to be about $110 billion, Regan said. He said a lower deficit for 1984 was "what we are going to shoot for, definitely." He added, "As far as I'm concerned, there should be no tax increase" in 1984, but then left open the option of advancing the Social Security payroll tax increases now scheduled for later years.

On unemployment, Regan said economists told him that even if the economy recovers, the underlying, structural level of unemployment is now lodged as high as 6 to 6 1/2 percent. This echoes comments made last week by the president's nominee for the chairmanship of the Council of Economic Advisers, Martin Feldstein. Regan said he hoped and believed that the economy would reach this "full employment" level in three to five years. Feldstein gave President Reagan the latest of the regular presidential briefings on the state of the economy, Regan said.

The recent decline in interest rates -- which yesterday included a move to 13 percent in the National Bank of Washington prime rate -- will help spur the economy, Regan said, adding the Federal Reserve Board has provided "plenty of money."

Meanwhile yesterday, Reps. Jack Kemp (R-N.Y.) and Trent Lott (R-Miss.) introduced a bill to "end the monetarist experiment begun in October of 1979" and to target interest rates. Democrats also have introduced House and Senate measures designed to make the Fed keep interest rates down to normal historic levels after accounting for inflation.