The National Labor Relations Board’s funding would be cut 12 percent under a White House budget proposal that asks lawmakers to fund the agency at $242 million for fiscal year 2020.

President Donald Trump’s administration has requested slightly less money for the agency since 2016, though lawmakers have kept the board’s budget flat at $274 million in each of the previous six years. The White House previously requested a 9 percent cut to the board’s budget that was rejected by Congress last year.

The NLRB is likely to remain funded at the same rates, or end up with a budget...