Labor leaders are nervous about Donald Trump’s appeal to unions’ many white, working-class members, and they are working to head off partisan defections.

Unions spend heavily to support Democrats in elections and wield great influence over whether their members support those candidates. But labor leaders fear many of their members could be drawn to Mr. Trump. Merged Wall Street Journal/NBC News polling data from the first four months of the year show that among white union households, support is split evenly between Mr. Trump...