WASHINGTON—President Donald Trump’s trade team this week will begin the complicated task of rewriting the sprawling North American Free Trade Agreement, the 23-year-old pact he has branded a “disaster” for U.S. factory workers. The negotiators’ challenge: finding a way to address workers’ concerns while keeping the promise to “do no harm” to the large number of industries and communities that have benefited.

The negotiations with Mexico and Canada, starting Wednesday in Washington, mark Mr. Trump’s first concrete attempt...