The Labor Department is wiping an Obama-era rule off the books that sought to give labor unions and workers more insight into talks employers have with legal counsel about thwarting union organizing campaigns.

The department rescinded on Tuesday a 2016 regulation known as the “Persuader Rule,” which was rolled back because it exceeded the department’s authority under the law and impinged on attorney-client privilege by requiring the disclosure of confidential information, the Labor Department said.

“By...