WASHINGTON—Ten days after his inauguration, President Donald Trump promised to “do a big number” on the Dodd-Frank law that tightened rules on financial firms after the 2008 crisis.

Behind the scenes, his then top economic adviser and a powerful senator settled on a less ambitious plan. And in recent weeks, Mr. Trump called a senior House lawmaker, urging him to move forward despite objections from Republicans who wanted broader changes.

The...