Initial claims for state unemployment benefits climbed to 603,000 in the week ended May 22, only the fifth time since the recession began last July that the 600,000 level has been breached, the Labor Department reported yesterday.

The high level of claims indicates layoffs are continuing in many industries, but workers are being rehired in significant numbers, too.

Analysts are uncertain how these two trends were balanced last month. However, they generally expect the national unemployment rate for May, which will be released this morning, to show at least a small increase from April's 9.4 percent figure.

The bad news on claims, which were up 25,000 from the seasonally adjusted level of 578,000 in the previous week, was offset somewhat by signs of rising consumer confidence that may be signalling a consumer-led recovery later this year.

In the monthly survey conducted by the Conference Board, a business research group, the board's consumer confidence index rose to 58.3 last month, up four points from April and five points from March. In addition, the proportion of respondents describing business conditions as "bad" fell from 54 percent in April to 48 percent last month. Nearly one-fourth expect the economy to improve in the next six months, the board said.

"The improvement in consumer expectations lends credibility to the prevailing professional view that business will begin recovering from the present recession in the months immediately ahead," said Fabian Linden, the board's director of consumer research. The gains in the index--in which the confidence levels of 1969-70 equal 100--have been strong enough that they may be indicating "that the consumer could lead a more robust expansion than is being projected by the economists," he added.

Meanwhile, several major retail chains reported modest sales gains in May over a year ago. Sears, Roebuck and Co., the nation's largest, said its sales were up 7.2 percent from May 1981, and some other chains had even larger increases.

And the three major automobile makers said yesterday that sales were up 10 1/2 percent in the final 10 days of May, and up almost 15 percent for the whole month compared with the same periods a year ago.

Auto sales have been depressed by several factors, including high interest rates. Yesterday First National Bank of Boston became the second among the nation's 20 largest banks to cut its prime lending rate from 16 1/2 percent to 16 percent. Citibank, the second largest, cut its prime last week, but other banks have been reluctant to follow because of an uncertain outlook for their cost of obtaining funds to lend.

The Labor Department also reported an increase in the number of persons receiving regular extended unemployment benefits from 588,500 to 597,100 for the week ended May 15. Under this program--which is available in 31 states with high unemployment levels--benefits are paid for an additional 13 weeks once payments under the basic 26-week program run out. The department said 37,500 persons exhausted their extended benefit eligibility in the latest reported week. None of the figures for extended benefits is seasonally adjusted.