WASHINGTON—A day after President Trump threatened to impose across-the-board tariffs on Mexico if it doesn’t crack down on the flow of migrants into the U.S., business groups and lawmakers expressed opposition to the plan, while the Mexican government said it would meet with the U.S. to explore a diplomatic solution.

Mr. Trump’s threat late Thursday that the U.S. could impose 5% tariffs on nearly $360 billion in Mexican imports was known in advance by just a small group, and the plan was opposed by his top trade adviser, Robert...