Construction will begin next month on a 348-unit, high-rise apartment on South Eads Street near the Pentagon City Quality Inn and Shirley Highway in Arlington. The developers are Edward H. Palmer of Chicago and Huntoon, Paige Associates of New York City. Rents will range from $402 for an efficiency to $616 for two bed-rooms, with utilities included. Broker Wallace B. Agnew handled the $1.6 million sale of the three-acre site, owned by John Stone. A $13.5 million package of construction and permanent financing for the HUD-insured loan was arranged by the Reilly Mortgage Group. McColl Construction of Chicago and Mechanical Contractors Inc. of Alexandria will handle the general construction jointly.

Housing Data Reports Inc. reported that the average price of new single houses, town houses and multiplex dwellings in area subdivisions rose 6.3 percent to $78,150 from Feb. 1 to Aug. 1. The increases in Montgomery County were the highest. Generally, the gap between new town houses and new detached houses in the area is now about $25,000.

Residential building permits in the District in August totaled 661 - 109 more than a year earlier - for a composite $13.6 million in new construction. Permits were issued for semi-detached brick dwellings in the 7400 block of 9th Street N.W., new brick flats in the 1900 block New Hampshire Avenue NW and new town houses over town houses at Sutton Place in the New Mexico Avenue area of NW, among others. Total residential permits issued this year in D.C. amount to 1,683.

In August, 67 houses were sold through the new multiple listing service for the District set up by the Washington Board or Realtors, the board reported. The average sale price was $106,700. At the end of the month, there were 425 active listings for houses in the city, the board said.

Boxing champ Muhammad Ali recently sold a three-story office building at 9001 Braddock Rd, Springfield, to Braddock Building Corp. for $1.7 million. Marie McQueen of Mount Vernon Realty handled the transaction. Attorney Daniel Clemente, who represents Ali in real estate transactions here, said that the boxer bought the office building in December 1976.