WASHINGTON—One-third of middle-class American adults couldn’t afford a $400 surprise expense, and some 6% also couldn’t manage such a cost even by borrowing money or selling something, a soon-to-publish Federal Reserve survey of household economics is expected to conclude.

Fed governor Lael Brainard said in a speech Friday that the Fed’s 2018 Survey of Household Economics and Decisionmaking, expected to be released later this month, would show that while economic security has been “marginally” improving, many U.S. citizens—both...