Three of the District's biggest and most prestigious law firms are planning to open offices in Fairfax County by early next year, the clearest signal of the county's growing attraction for corporate lawyers seeking to capitalize on its robust economy.

Covington & Burling, the city's largest firm, will open its first office outside the District early next year somewhere in the Tysons Corner area, according to partner Edward Dunkelberger.

Hogan & Hartson, which ranked as the city's third largest in a recent listing by the National Law Journal, will open a six-lawyer office in Tysons Corner late this year, and it hopes to have 20 lawyers working there within a year, partner Richard J.M. Poulson said.

And Arent, Fox, Kintner, Plotkin & Kahn, which ranked sixth in the Law Journal survey, will open in Tysons Corner late this year or early next year with five to seven lawyers, including former Virginia attorney general J. Marshall Coleman, who joined the firm last month, according to partner Rodney F. Page. Page, a former chairman of the Fairfax County School Board, said the firm is aiming to have 10 to 12 lawyers in the Tysons Corner office within 18 months.

"Attorneys and law firms follow economic growth," said Rosalie Small, executive director of the Fairfax Bar Association. "Where there's construction, where there's companies moving in, where there's high-tech industry, the lawyers and the law firms are going to follow." Membership in the Fairfax Bar Association, she said, has grown from 450 in 1977 to nearly 1,300.

Plans by the three firms to move follow the mergers during the past 16 months of three large Baltimore firms with firms in Fairfax, and the opening of a branch office in Fairfax by Richmond's Hunton & Williams, the biggest law firm in the state.

Another major District firm, Shaw, Pittman, Potts & Trowbridge, set up a six-lawyer branch office in McLean last year, and it plans to have more than 20 lawyers practicing in Fairfax within a few years. Pittsburgh's biggest firm, Reed, Smith, Shaw & McClay opened a McLean outpost in 1981.

Page of Arent Fox said, "For years we have watched what was going on in the suburbs." Now, he said, "The Washington region is growing so large, it's only logical to think there's not a single downtown where people are going to go for business services but several."

The firms' decision to locate in Tysons Corner cements a shift in the center of legal activity in the county away from the courthouse in Fairfax City to the Tysons area, where business construction is booming. The law firms, which specialize in litigation and legal advice for corporations, hope to be frequently called-on neighbors.

"The feeling is now that there are maybe a number of companies, new-entry companies, particularly in the high-technology area, who are not oriented to Washington, D.C., and who might well value a law firm working out there," said Dunkelberger of Covington & Burling.

While some Covington clients are already located in Fairfax, Dunkelberger said, "Also, quite frankly, this is sort of a forward-looking thing: 'Let's see if Northern Virginia is the new Silicon Valley.' "

The numbers suggest that it may be. During fiscal year 1985, more than 240 companies announced plans to open new facilities or expand existing operations in Fairfax County, according to the Fairfax County Economic Development Authority. The total investment by those firms will exceed $818 million and bring 32,500 new jobs to the county over the next five years, the authority estimates.

"It's clear now that this is a legitimate commercial center," said C. Thomas Hicks III of Shaw Pittman. "The clients are here, and many of them would prefer not to have to go downtown" to meet with their lawyers.

As the county's business booms, lawyers said, the firms are filling an accompanying need not just for more legal help, but also for advice of a more sophisticated and specialized nature than the more nuts-and-bolts legal guidance Fairfax firms have traditionally offered.

"There is a recognition by existing firms that there is not a suffficient level of expertise to service these clients," said William D. Dolan. Dolan's five-lawyer firm, Dolan, Treanor, Murray & Walsh of Arlington, merged last February with a large Baltimore firm, Venable, Baetjer & Howard, opened a six-lawyer office in Tysons Corner last August, and it expects to have a total of 20 lawyers in its two Virginia offices by August 1986.

"Very sophisticated tax advice, health law expertise, labor expertise, international law -- all the things that have traditionally been done solely by major firms buried in the heart of the urban area in downtown Washington, they are now being done in what was yesterday a suburb," Dolan said.

The companies in Fairfax, said Poulson of Hogan & Hartson, "are doing business with the government and have the same sorts of problems that firms like ours are used to handling."

In another signal of the growing importance of Tysons Corner as a legal center for the county, the largest law firm in Fairfax, 65-lawyer Boothe, Prichard & Dudley, moved to Tysons this month from Fairfax City.

"We now regard Tysons as really the downtown of Fairfax County," said R. Dennis McArver, managing partner of the 93-year-old firm. "More and more clients of ours seem to be located in this area."

McArver said the firm is maintaining only a "rear guard" office in Fairfax City staffed by one or two attorneys.

"The Tysons area is tailor-made for our firm," said Bob Odle of Hogan & Hartson, noting that Tysons has more office space than Richmond and Norfolk combined. "We feel there are some very substantial businesses located in the Tysons area and we could more conveniently offer our services to them if we had an adequately staffed office out there," added Odle, who compared Hogan's planned Tysons outpost to a midtown Manhattan office of a Wall Street law firm.

The growth of the county, meanwhile, has made Fairfax "a different place to practice law," said William L. Stauffer Jr., whose six-lawyer Stauffer & Dorn merged in June with 112-lawyer Frank, Bernstein, Conaway & Golden of Baltimore.

"The small lending institutions we used to represent are now major state banks. The small businesses, some of them have grown up to be big businesses." said Randolph W. Church, managing partner of the Fairfax office of Hunton & Williams. "It's changed from essentially almost a country law practice, a bedroom community law practice, to what's very similar to a metropolitan law practice."