A lawyer for Donald Trump Jr., President Donald Trump’s eldest son, on Tuesday asked the House Intelligence Committee to open a “formal inquiry” into what he said were leaks by committee members or staff following the younger Mr. Trump’s interview with the panel earlier this month.

The lawyer, Alan Futerfas, in a letter sent to House Intelligence Committee Chairman Rep. Mike Conaway (R., Texas), asked the panel to investigate in particular “whether any members or staff members of this committee intentionally disseminated inaccurate...