Former Trump adviser Jason Miller said Mr. Trump and his allies are seeking to make impeachment “politically painful for Democratic freshmen” and argued the pitch could resonate with independent voters. “I do believe there’s an opening here to make a point we can win with independents,” he said. “They hate politics, they hate the machinery.”

But Mr. Miller also echoed the concerns of many Republicans that the White House and the campaign needed to get behind a more unified strategy, after weeks of reacting to the president’s tweets. Others wondered whether issues such as jobs and the economy would get lost and if the combative messaging would drown out the campaign’s outreach efforts to women and minorities in recent months.

Meanwhile the campaign and the RNC have been dropping millions in freshly raised dollars on campaign ads targeting House Democrats who hold districts won by Mr. Trump in 2016. Their messages echoed Mr. Trump’s declaration that the investigations were a “witch hunt” and a “coup.”