The Federal Trade Commission has endorsed a roughly $5 billion settlement with Facebook Inc. over a long-running probe into the tech giant’s privacy missteps, according to people familiar with the matter.

FTC commissioners this past week voted 3-2 in favor of the agreement, with the Republican majority backing the pact while Democratic commissioners objected, the people said. The matter has been moved to the Justice Department’s civil division and it is unclear how long it will take to finalize, one of the people said. Justice...