A spreading wave of recession layoffs pushed the nation's unemployment rate to 6 percent in July at the same time wholesale prices were taking another sharp leap upward, the Labor Department said yesterday.

To go with that bad news, major banks said they were raising their prime lending rate to a record 12.75 percent. [Story, Page D9.]

The reports underscored the problem now facing the Carter administration as it tries to combat a deteriorating economy while slowing down the rate of inflation.

The unemployment rate rose from 5.7 percent to 6.00 percent in August, the highest rate in a year. The number of persons looking for work jumped to 6,149,000.

On the inflation front, a new surge in food prices and continued increases in energy prices pushed the producer price index for finished goods up 1.2 percent, or an annual rate of 15.4 percent.

The rise in unemployment prompted calls from several members of Congress for a quick tax cut. "I've said it before and today's unemployment figures compel me to say it again," declared Sen. Lloyd Bentsen (D-Tex.), chairman of the Joint Economic Committee. "We need a tax cut, and continued delay in facing up to this need will only cause more people to be thrown out of work."

President Carter recently directed his economic advisers not to discuss the possibility of a tax cut, and sources say his instruction is being followed to the letter, even in private meetings of his Economic Policy Group.

Janet Norwood, commissioner of labor statistics, told the JEC that the August labor market information "reinforces the signals of weakening economic performance from many other statistical series." Not only are layoffs rising, she added, but the number of hours being worked is also dropping.

Producer prices for consumer foods rose 1.2 percent last month after four months of declining or stable food prices. Gasoline prices soared 6.1 percent compared to 3.9 percent in July. Heating oil prices climbed a full 6 percent, too, but that was less than July's 9 percent.

With prices continuing "to escalate in the double-digit range," Norwood testified, "real wages declined."

The increases in prices charged by producers -- the Producer Price Index has replaced the old Wholesale Price Index -- "suggest that no substantial improvement in consumer goods [prices] occurred in August," she said. Consumer prices rose at a 12.7 percent annual rate in July. The August CPI will be released later this month.

Total employment fell by 310,000 in August to 96.9 million. The total number of jobs has shown no growth since March, the department reported.

So far, the rise in unemployment has affected different demographic groups unevenly. The unemployment rate for adult men was little changed at 4.2 percent, while the rate for adult women rose from 5.5 percent in July to 5.9 percent in August. The rate for teen-agers climbed from 15.3 percent to 16.5 percent.

The rate for blacks for virtually unchanged at 11 percent. The rate for white workers, on the other hand, rose from 4.9 percent to 5.3 percent.

Layoffs were concentrated in manufacturing and construction, and there were also drops on jobs in primary and fabricated metals. Many of the layoffs could be attributed to cutbacks in automobile production, but because of seasonal adjustment problems related to annual model year changeovers, the department was unsure about their magnitude.

While factory jobs are shrinking, employment in the service-producing sector is still going up -- by 155,000 in August alone.

Average hourly earnings of production or nonsupervisory workers on private nonagricultural payrolls rose 0.2 percent in August and were 8 percent above a year ago. However, consumer prices were up 11.3 percent in the 12 months ended in July. The hourly earnings index, which is adjusted for changes in overtime and in the proportion of workers in high-wage and low-wage jobs, rose still less, 7.6 percent.

There were some signs in the Producer Price Index for intermediate goods and for crude goods that inflation might be slowing. Norwood said, however, that the improvement "occurred at earlier stages of production which, if sustained, will take several months before they are reflected in the Consumer Price Index."

Those improvements included only a 0.1 percent rise in the index for crude materials, and a similar increase in prices for capital equipment bought by business that was the smallest one-month change in six years, the department said.

Bentsen again proposed a $20 billion tax cut that would include reductions for business as an incentive for new investments that, he said, could lead to lower rates of inflation in the future.

"I realize that many of my colleagues have been hesitant to support a tax cut because we face not only unemployment but inflation as well," he said, adding, "I'm talking about supply side tax cuts that will help combat the current recession and also help moderate inflation by boosting productivity."

Bentsen also took issue with administration officials, including Treasury Secretary G. William Miller, who have been saying that it is too early to propose a tax cut and that the recession may be so mild that none is needed.

"We must not repeat the mistakes of the 1974 recession," Bentsen declared. "In that case a tax cut was enacted later, after much foot dragging and indecision . . . Late timing on that tax cut unnecessarily cost a lot of Americans their jobs."

Miller, this week, said that the recession may already be half over "and we haven't had tremendous strains or pressures." On the other hand, yesterday's report of a 300,000 rise in the number of people unemployed was the first unmistakable sign that the recession was hitting the job market. "Strains and pressures" are expected to mount as the unemployment rate continues to rise.

Officially, the administration has predicted that the rate will reach 6.9 percent late next year. A number of administration economists believe it could top 8 percent by a year from now.