WASHINGTON—The White House on Monday took to television, social media and email to discredit accusations from former national security adviser John Bolton that threatened its defense of President Trump. But inside the Senate chamber, where the president’s team was making the official case for why he should remain in office, there was scant mention of Mr. Bolton—or his book.

The double-barreled strategy underscored the challenge facing Mr. Trump’s impeachment team of satisfying both a jury of senators—who are days away from...