The Federal Reserve Board yesterday approved plans for investors from Saudi Arabia, Abu Dhabi and Kuwait to buy Financial General Bankshares Inc., the holding company that owns the First American Banks and others in Virginia, Tennessee and New York.

The Middle Eastern investors now need only the approval of New York State banking regulators to complete a bid for control of Financial General that began three and one-half years ago.

If New York approval is obtained as expected, the Middle Eastern group will make a public offer this fall to buy all of Financial General's outstanding stock, said their attorney, Robert Altman, partner in the Washington law firm of Clifford and Warnke.

Altman said his clients plan to spend about $220 million on Financial General, including money they used in 1978 to buy 19 percent of the bank's stock, the cost of the remaining shares, and additional capital that will be invested to enable the banks to expand.

The application approved by the Federal Reserve Board calls for the Middle Eastern group to provide $12 million in new capital to pay for additional bank branches and establishment of a new international banking department for the New York bank.

The buyers of Financial General include Sheik Kamal Adham, a Saudi Arabian businessman who once headed that country's central intelligence agency; Faisal Saud al Fulaig, former head of Kuwait's national airline; and Abdullah Darwaish, financial manager for the royal family of Abu Dhabi.

The group has set up a chain of companies incorporated in the United States, The Netherlands and the Netherlands Antilles to buy Financial General. The actual purchase will be made by FGB Holding Corp., which recently was organized in Delaware. FGB Holding is owned by Credit and Commerce American Investment of Holland and Credit and Commerce American Holdings of the Netherlands Antilles.

Altman said "the investors are deeply gratified by the board's decision" that followed "a lengthy, comprehensive examination of the proposed acquisition."

Financial General officials issued a statement noting the Federal Reserve Board action and reminding stockholders that approval of New York regulators is needed before the takeover can be completed.

Under an agreement signed last year, the Middle Eastern group promised to pay 1.58333 times the book value of Financial General's common stock for all outstanding common shares. The bid was worth $28.50 a share at the time it was made and calls for a price of $31.96 a share based on the June 30 value of the shares.

Beside the Middle Eastern group, the biggest stockholders of Financial General are Washington real estate man B.F. Saul II, who is now chairman of the board, and retired Rockville investor Eugene B. Casey, a long-time board member.

Casey long urged Financial General officials to make a deal with the Middle Eastern investors. Saul led a lengthy fight against the takeover that took more than three years but boosted the bid for the company's stock from the $15 a share originally offered to more than twice that amount.

Saul and other Financial General executives finally agreed to the takeover in May of 1980. Since then, the buyers have been getting the purchase approved by state and federal regulators.

Altman said the final application for New York State approval "is proceeding routinely" and probably will be completed this fall.

With assets of $2.3 billion, Financial General is the third-largest banking company based in the District of Columbia, after Riggs National Bank and American Security Corp. Financial General's First American banks hold about 10.2 percent of the deposits of D.C. banks, 4.7 percent of Virginia deposits and 2.2 percent of Maryland business.

Besides the First American banks in the Washington area, Financial General owns Peoples National Bank of Leesburg, Shenandoah Valley National Bank of Winchester, Valley National Bank of Harrisonburg, Round Hill National Bank of Round Hill, Eastern Shore National Bank of Pocomoke City, Md. Bank of Commerce in New York City, Community State Bank of Albany, N.Y., and Valley Fidelity Bank and Trust Co. of Knoxville, Tenn.

The Middle Eastern buyers told the Federal Reserve they plan to hire an experienced American banker as chief executive of Financial General. FG President J. William Middendorf II resigned recently to become ambassador to the Organization of American States.