Agency Bank. A separately incorporated branch of a foreign bank inside the United States. Agency banks cannot make loans or take deposits in their own name; rather, they do so on behalf of the parent bank in the foreign country.They also are the main issuers of American Depository Receipts.gency Bank. A separately incorporated branch of a foreign bank inside the United States. Agency banks cannot make loans or take deposits in their own name; rather, they do so on behalf of the parent bank in the foreign country.