The housing market in the Bay Area remained strong in May, with record prices posted in Santa Clara and Alameda counties, but the rise in home prices and sales has begun to level off, according to a report released Thursday.

“Instead of getting 20 offers on a house, sellers are getting, three, four, five offers,” said Mike James, president of the Concord-based Bay Area office of Coldwell Banker, a realty firm. “But there is enough pent-up demand that I wouldn’t anticipate a major market shift.”

Home prices rose at double-digit levels in four Bay Area counties in May compared with the same month a year ago, according to CoreLogic, which tracks the real estate market. The figures were for resale activity involving single-family homes.

The median sale price in Santa Clara County hit a record $900,000, CoreLogic said.

“In terms of generating demand for housing, Santa Clara County has more going for it than the rest of the country when it comes to job growth and wage growth,” said Andrew LaPage, a research analyst with CoreLogic.

Alameda County home sales hit a record price of $708,000 in May. San Francisco reported a median price of $1.13 million that month.

The median sale price for all previously owned homes sold in the Bay Area in May was $700,000, CoreLogic reported. That was up 6.9 percent from the year before.

Sales activity in the Bay Area in May was 5.5 percent higher than the same month a year ago.

“Demand for housing remains healthy in the Bay Area,” LaPage said.

In Contra Costa County, the median home price was $485,000, up 2.1 percent from the year before, while sales in that county were up 7.4 percent over the past 12 months.

Double-digit increases in home prices occurred in San Mateo County, up 18.1 percent; San Francisco, up 17.5 percent; Sonoma County, up 11.4 percent; and Santa Clara County, up 10.4 percent.

“The job market justifies these kinds of increases in the Bay Area,” James said. “Tech companies are doing very well. All of a sudden you hear more about startup companies. There is still pent-up demand for housing.”

Santa Clara County has the hottest job market — by far — in the entire country.

Over the 12 months that ended in April, the total number of payroll jobs in Santa Clara County grew by 6 percent, according to the U.S. Bureau of Labor Statistics.

The San Francisco-San Mateo County region was No. 2 in the country, although a distant second. That region posted a 4.9 percent increase in annual job growth over the 12 months that ended in April. The East Bay posted a 2.1 percent gain in total jobs in that period, the government reported.

Still, while sales and price increases remain higher and are hitting record values in some areas, the rate of the increase is at a reduced pace for both sales and prices.

“You are still running into inventory hurdles in some places as well as affordability and credit constraints,” LaPage said. “That is what is limiting the amount of activity.”

The leveling of activity means sellers can’t simply price homes $200,000 or $300,000 above current market prices for an area.

“Sellers who are reaching for the sky, the buyers are saying ‘wait a minute’ when they see homes listed at well above the market,” said David Tonna, a past president of the Silicon Valley Association of Realtors and a broker associate with Alain Pinel Realtors in Los Gatos.

Contact George Avalos at 408-859-5167. Follow him at Twitter.com/georgeavalos.