First American Bankshares Inc., a Washington-based bank holding company, sold its majority interest in Bank of Commerce of New York City yesterday to Norstar Bancorp Inc. of Albany, N.Y., for $34.4 million.

In a separate transaction also announced yesterday, First American Bankshares said it has proposed a tender offer for all of the common stock of Valley Fidelity Bank & Trust Co. of Knoxville at $64 a share. First American Bankshares now owns 36.6 percent of the stock outstanding in Valley Fidelity. First American would pay about $20 million in a transaction that would give it control of the Tennessee bank.

First American Bankshares is a Virginia corporation with headquarters in the District. In addition to its three First American Banks in the Washington area, it has 11 banks, principally in Virginia, Maryland and New York.

It assigned an option earlier this year to Norstar Bancorp, a $3.6 billion Albany bank holding company, to buy First American's interest in Bank of Commerce.

Bank of Commerce and Community State Bank of Albany were two of the 11 banks that had been owned by Financial General Bankshares Inc., a multibank holding company. Financial General was renamed First American Bankshares last year after a group of Middle Eastern investors bought control.

New York state banking authorities, like their counterparts in other states where Financial General owned banks, had objected initially to the company's takeover by Arab investors. New York authorities subsequently approved an application by the group to form a bank holding company to acquire Financial General, however.

Approval of the application in New York followed a compromise in which the Middle Eastern investors agreed to sell Financial General's 67 percent interest in Bank of Commerce. In return, Financial General--now First American--was allowed to maintain ownership of Community State Bank of Albany.

First American said yesterday it will soon open a branch of the Albany bank in New York City, where it plans a significant expansion. Funds received from the sale of Bank of Commerce will be used for development of the new bank, which will be called First American Bank of New York.

First American Bankshares has been a major stockholder in Valley Fidelity Bank & Trust Co. of Knoxville and its predecessors since 1916.