First American Bankshares, a Washington-based holding company with a majority interest in 11 banks, has initiated steps to merge five small Virginia banks into the much larger First American Bank of Virginia.

The merger proposal is part of a three-phase plan designed to make First American of Virginia a statewide banking institution, with nearly 100 branches and $2 billion in assets. McLean-based First American operated 45 branches and had assets totaling $1.1 billion at the end of 1982.

The five First American Bankshares banks that would be merged into First American of Virginia are: Shenandoah Valley National Bank of Winchester, Valley National Bank of Harrisonburg, The First National Bank of Lexington, The Peoples National Bank of Leesburg and the Round Hill National Bank.

Under the merger plan that has been proposed, the stock of each subsidiary bank would be exchanged for stock in First American of Virginia. If approved by regulatory authorities and directors and shareholders of the banks involved, the mergers would not be completed until the end of the year, First American Bankshares said.

The company also said that management and staffs of the banks would be retained in an effort to stress local business and consumer services. At the same time, directors of the subsidiary banks would be retained as regional directors in a statewide banking network.

The mergers and acquisitions would result in "significantly better service" for customers of the subsidiary banks, said Robert G. Stevens, president of First American Bankshares. As examples of improved services, he noted, local lending authority would be increased and customers would be able to make deposits and withdrawals at any First American office in the state.

The First American banks were the first to be linked in all three Washington jurisdictions by 24-hour automatic teller machines. The ATMs, called Money Exchange machines, will be installed throughout the expanded market when the mergers are completed.

By consolidating its Virginia banks, First American Bankshares is proceeding with a strategic plan that had been expected to be implemented after the holding company, formerly Financial General Bankshares Inc., was bought by Middle Eastern investors last year.

The core of First American Bankshares' banking operations is its three First American Banks in Virginia, the District and Maryland. The company announced to shareholders earlier this year that it was embarking on a strategic program to develop the business of its three major banks in the Washington region.

Under a separate plan disclosed this past weekend, First American Bank of Virginia has also entered into a merger agreement with the First Peninsula Bank & Trust Co. of Hampton, Va.

In the meantime, First American Bank is in the process of concluding an agreement to acquire 26 branches of Virginia National Bank of Norfolk and First & Merchants National Bank of Richmond. Virginia National and First & Merchants were forced to put the branches up for sale after agreeing to merge earlier this year.

The mergers and branch acquisitions would extend First American's market area from Northern Virginia to the Shenandoah Valley and the Tidewater area of the state.

In addition to its Washington-area First American Banks, the $2.4 billion First American Bankshares owns a majority interest in First American bank of New York; Eastern Shore National Bank of Pocomoke City, Md., and the Valley Fidelity Bank & Trust Co. of Knoxville, Tenn.