Raising the possibility of a bidding contest for control of the Washington area's largest financial institution, directors of Riggs National Bank yesterday formally rejected financier Joe L. Allbritton's bid to buy a controlling interest in the bank. They urged stockholders to turn down Allbritton's offer, calling it "inadequate."

Riggs said its management has approved discussions of alternative bids with "selected financial institutions." This suggests that current stockholders might get a higher offer from rival investors as friendly by Riggs management, adding to the dilemma of bank stockholders who face an intitial deadline of tonight to respond to the Allbritton bid.

Riggs also revealed yesterday that it has hired two firms to help in its defense against Allbritton. The investment firm of Keefe, Bruyette & Woods of New York, which specializes in banking companies, has been retained for investment banking services and already has issued an opinion that the Allbritton offer of $67.50 a share "is inadequate" based on "the excellent condition of the bank, its fine operating record, the relationship of the price offered to the book value [$59 a share at the end of 1980] and underlying earnings . . ."

In addition, Riggs has hired a New York firm that specializes in proxy solicitation, D.F. King & Co., which portends a battle for the minds and pocketbooks of Riggs stockholders.

In a letter mailed to stockholkers last night and in a petition filed with the comptroller of the currency late yesterday, Riggs Chairman Vincent Burke Jr. and President Daniel Callahan III said the Allbritton offer last week of $67.50 apiece for at least 600,000 shares "is not an adequate price to pay for control of a bank with Riggs' record and prospects."

Burke and Callahan thus ignited a direct confrontation between one of the nation's wealthiest men and the Washington establishment, as represented by the historic bank' 25 director's.

If there were any doubt about animosity between the board and Allbritton before the documents were made public yesterday, the letter to stockholders makes clear the board's opposition to Allbritton himself. The Riggs officers expressed concern about his attempt to gain "veto power" over a bank holding plan that he opposed and raised questions about heavy interest costs he must bear on $70 million of bank loans to buy Riggs stock.

Noting that the board voted in January that it was not in the "best interests" of the bank for any person to own more than 15 percent of its stock, Burke and Callahan went on to say that "in light of the Allbritton offer . . . this policy, while desirable, may not be one the bank can now realistically pursue."

That indicates that the Riggs directors might approve a rival bid for control from investors other than Allbritton, a former Texas banker whose business interests include newspapers, television stations, insurance and mortuaries. Many long-time business leaders here still consider Allbritton an outsider even though he has a home here and has established Allbritton Communications in the District.

When the January board decision was revealed, banking and investment sources here translated it to mean that directors of the bank specifically opposed Allbritton buying any shared than the 15 percent he now owns. Informed sources said he was angered by this reaction to what he saw at the time only as a major investment and decided to move quickly with a stock offer to gain control.

Some directors, meanwhile, have been said to question whether Allbritton is seeking a long-time investment in an institution known for its stability of if he is just seeking a quick profit, such as the gains he made first by buying and then selling The Washington Star in the 1970s -- but retaining lucrative Star broadcast properties.

In their letter last night, for example, Burke and Callahan said the board is concerned that "Allbritton has included as one of his future options a decision to dispose of his holdings in the bank'," according to his formal offer.

Racing against an initial deadline of tonight in Allbritton's complex bid to buy at least 600,000 shares of Riggs stock at $67.50 apiece, a $42.75 million deal that would increase to 35 percent his ownership in Riggs the 25 Riggs directors met last week and again yesterday morning before adopting the strategy revealed last night.

In his surprise bid last week, Allbritton said that if 600,000 or fewer shares were tendered by midnight today and not withdrawn by March 3, he would buy those shares. The Allbritton offering is due to expire March 10 unless extented by him or by potential federal regulatory inquiries into the acquisition.

Riggs officers said yesterday the board has approved discussion by financial advisers and management with selected financial institution "that may have an interest in exploring alternatives to the Allbritton offer."