Wisconsin Gov. Scott Walker, who rose to national prominence by battling his state’s public-sector labor unions, is set to unveil Monday a plan to strip power from unions nationwide.

The Republican presidential candidate’s proposal, which he plans to announce at an afternoon speech in Las Vegas, would eliminate the National Labor Relations Board, prohibit federal employee unions, institute right-to-work laws nationwide and repeal the Davis-Bacon Act of 1931, which requires the payment of local prevailing wages to workers on...