The Middle Eastern investors trying to buy control of Financial General Bankshares Inc. are looking for an American banker to run the company, according to documents released yesterday by the Federal Reserve Board. by the Federal Reserve Board.

The person selected to run the company will be named "within the next few weeks," attorneys for the Middle Eastern group disclosed in a massive application fro permission to form a bank holding company.

Federal Reserve approval is the first barrier the group must clear before it can proceed with plans for a public tender offer for all Financial General common stock.

If the tender offer is successful, the application says, the new owners plan to replace Financial Gereral Chairman B. Francis Saul and President J. William Middendorf.

A letter in the nine-inch-thick file indicates the application won't be acted upon until the Fed finds out who will be running the bank.

Attorney Robert Altman, who filed the application, said, "the intent would be to get an American who is well versed in banking" for the job.

The only American banker associated with the group so far has been President Carter's former budget director, Bert Lance, who helped the Middle Eastern investors buy about 18 percent of Financial General's stock.

But Lance is no longer involved in the takeover group, the application says, and will not be involved in management of the bank. Lance's activities as a Georgia banker before he came to Washington are still being investigated by a federal grand jury in Atlanta.

Replacing Lance as the leading American representative of the Middle Eastern group is former Democratic Sen. Stuart L. Symington of Missouri.

Symington will serve as trustee for the stock in the new bank holding company that is owned by Sheik Kamal Adham of Saudi Arabia, and Faisal Saud al Fulaig of Kuwait.

The application spells out plans of the takeover group to centralize the management of the Financial General banks, which now operate with relative autonomy. Major banks in the chain include Union First National of Washington, First American Bank of Virginia and American Bank of Maryland.