More than half of Americans — and an overwhelming number of Democrats — say they approve of the inquiry, according to a CBS News poll released Sunday. But the survey found a partisan split, with most Democrats calling the president’s handling of Ukraine illegal and most Republicans calling Mr. Trump’s actions proper — or, if improper, at least legal.

The week-old inquiry is barreling forward, even with lawmakers out of town for a two-week recess. Mr. Schiff, appearing on the ABC program “This Week,” said Sunday that the whistle-blower who triggered the inquiry would testify “very soon.”

But Mr. Schiff hinted the committee might not call Mr. Giuliani, the bombastic former New York mayor who was essentially running a shadow foreign policy in Ukraine on behalf of Mr. Trump. Interviewed on ABC shortly before Mr. Schiff, Mr. Giuliani at first said he “wouldn’t cooperate with Adam Schiff,” then said he “will consider it.”

Sunday night was only the latest effort by Ms. Pelosi to try to strike a dignified tone for the process with her appearance on “60 Minutes.” In a series of interviews, she has been making the case that Mr. Trump engaged in “a cover-up,” calling this moment a “sad day for our country.”

Sign up for On Politics to get the latest election and politics news and insights. Sign up for our politics newsletter

But the carefully coordinated messaging campaign may be upended before it starts. Liberals are reveling in news of an inquiry that they believe should have been opened long ago. The campaign of Representative Rashida Tlaib, Democrat of Michigan, whose profane cry for impeachment made news on her first day in office, is already selling T-shirts emblazoned with the slogan using a two-letter abbreviation for the expletive she used back in January.

And on the presidential campaign trail, Democrats are talking up impeachment, which poses a danger that the public will think the party is prejudging the outcome of the inquiry and politicizing a solemn task that has grave implications for the future of the nation.

“We need to make sure this is fact-driven and evidence-based, “ said Representative Josh Gottheimer, a centrist Democrat from New Jersey who had resisted calls for the inquiry until now. “You can’t prejudge something that is so solemn and obviously could have a big historical impact on our country, and you need to keep the country together.”