The impasse over Mr. Trump’s proposed border wall and the death late last month of an 8-year-old boy from Guatemala in United States custody have catapulted the administration’s divisive policies at the border to the top of House Democrats’ agenda. Representative Jerrold Nadler of New York, the Judiciary Committee chairman, and a handful of other chairmen plan to examine immigration policies.

Their interest and detailed questions could prove problematic for Mr. Trump’s efforts to make a case for his border wall, shining a light on gaps in the current policy and trying to highlight the human toll on migrants.

“We are calling on D.H.S. to preserve any and all evidence related to these horrible incidents,” the chairmen wrote. “We as a country can and must do better.”

Mr. Nadler has already invited Matthew G. Whitaker, the acting attorney general, to be the committee’s first public hearing witness, and indicated in recent days that he would subpoena Mr. Whitaker if necessary to compel an appearance before the end of the month.