President Trump and his allies will fight impeachment in the halls of Congress and over the airwaves by attacking Democrats’ partisan motives, making their case particularly for white voters in battleground states that the president is a victim of “unhinged” liberal hatred, sources close to the president said.

The president huddled with more than a dozen House Republican lawmakers at the White House on Thursday, hours after House Democrats approved a measure to move forward with an impeachment investigation centered on Mr. Trump’s overtures to Ukraine’s leader to investigate 2020 Democratic presidential front-runner Joseph R. Biden.

“The Democrats are desperate,” Mr. Trump told a British interviewer after the vote. “They’re going to try and win the election this way, because they can’t win it the fair way.”

While Republicans can’t stop the impeachment probe, they can escalate their barrage of criticism that the investigation is illegitimate and fundamentally unfair, supporters of the president say. Regardless of a vote to impeach the president, the strategy — coming soon to TV — will help him in a handful of states that will decide the 2020 election, they say.

“Essentially it’s about a PR campaign that is meant for about 10 percent of the persuadable voters in a total of about six states — Florida, Pennsylvania, Wisconsin, Michigan, a little bit of Minnesota, North Carolina, Arizona,” Republican strategist Ford O’Connell said. “We’re really talking about a giant PR battle that’s going to carry over to the ballot box, more so than the actual trial itself.”

Sources familiar with the White House’s plans say the president’s team also is paying attention to the level of white voters in swing states who oppose Democrats’ push for impeachment.

“Two-thirds of the white voters do not want the president impeached,” said one GOP source. “The people in places like Florida, North Carolina, New Hampshire, New Mexico, Wisconsin, that are actually going to decide this election, they don’t want to see the president impeached.”

A New York Times Upshot/Siena College poll released Wednesday showed most voters across six battleground states are opposed to impeaching and removing Mr. Trump from office, even as they say they support House Democrats’ impeachment inquiry.

By a 52% to 44% margin, registered voters in Arizona, Florida, Michigan, North Carolina, Pennsylvania and Wisconsin said they oppose impeaching and removing Mr. Trump from office.

The Republicans’ strategy could include using televised hearings to turn the tables by putting on trial the leader of impeachment, Rep. Adam B. Schiff, California Democrat and chairman of the House Permanent Select Committee on Intelligence. Mr. Schiff also led House Democrats’ fruitless, long-running investigation into allegations of Russian collusion with the Trump campaign in 2016.

“Here’s my challenge to Mr. Schiff: You want to be Ken Starr? Be Ken Starr,” said Rep. Doug Collins, Georgia Republican, referring to the independent counsel who investigated President Bill Clinton. “Come to the Judiciary Committee, be the first witness and take every question asked of you, starting with your own involvement with the whistleblower. Folks, this ain’t over. Get ready.”

The whistleblower who initiated the case against the president contacted Mr. Schiff’s staff before filing a complaint accusing the president of improperly pressuring Ukraine’s president in a July 25 phone call to investigate Mr. Biden. The contact with Democratic staff raised accusations by the president and Republicans that Democrats’ probe was rigged from the start. Mr. Schiff has denied any improper contact.

“I think they’ll try to put Adam Schiff on trial,” a source close to the White House said. “But the president needs to let others do that: ‘When did you talk to the whistleblower? You lied to us.’ President Trump needs to focus on being president.”

Republicans’ well-funded campaign coffers also are coming into play. A Republican Super PAC released digital ads Thursday targeting 29 House Democrats who voted for the impeachment measure and represent districts that Mr. Trump won in 2016.

And Mr. Trump’s reelection campaign went on offense by running a 30-second ad Wednesday night during Game 7 of the World Series promoting the president’s political victories and attacking Democrats over the impeachment inquiry. The campaign spent more than $1 million on the ad buy, which President Obama’s former campaign manager David Plouffe called “unprecedented” so early in the campaign.

“He’s no Mr. Nice Guy, but sometimes it takes a Donald Trump to change Washington,” the ad stated.

Said Mr. O’Connell, “The fact that he’s an incumbent gives him advantages. The party apparatus is going to walk in lock step with him. And he’s got a lot of money. If he comes out on top, it could be his best reelection ad.”

The White House responded rapidly to the party-line vote in the House to move forward with the impeachment probe, calling it proof that Democrats are still fixated on overturning the 2016 election. White House press secretary Stephanie Grisham called it “a blatantly partisan attempt to destroy the president.”

“The president has done nothing wrong, and the Democrats know it,” Ms. Grisham said. “Nancy Pelosi and the Democrats’ unhinged obsession with this illegitimate impeachment proceeding does not hurt President Trump; it hurts the American people.”

The White House statement emphasized the cost to the nation in terms of unfinished legislative business, such as a trade agreement with Canada and Mexico, a plan for rebuilding infrastructure and a measure to lower prescription drug prices.

The White House also is focusing on the unfair process that Democrats have set up to prosecute the president.

“Speaker Pelosi, Chairman Schiff and the Democrats conducted secret, behind-closed-door meetings, blocked the administration from participating, and have now voted to authorize a second round of hearings that still fails to provide any due process whatsoever to the administration,” Ms. Grisham said. “The Democrats want to render a verdict without giving the administration a chance to mount a defense. That is unfair, unconstitutional, and fundamentally un-American.”

Some conservatives are raising alarms about what they view as the White House’s ineffective pushback so far to the threat from Democrats.

Jack Goldsmith, a former assistant attorney general in the George W. Bush administration and a senior fellow at the conservative Hoover Institution, noted that six current administration officials have answered lawmakers’ questions behind closed doors despite White House orders not to do so.

“This is a really remarkable breakdown of soft and hard presidential power,” Mr. Goldsmith said Thursday on Twitter, adding that the White House “isn’t even putting up a fight.”

“The WH has no juice, no tools,” he tweeted. “In so, so many ways, Trump is a weak, not a strong, president.”

Mr. O’Connell said there was another piece of “great news” hidden in the House party-line vote: even the dozen or so “Never Trump” Republican lawmakers in the House voted with the president.

“The rules that were set up by Pelosi and Schiff were so bad, you even got the ‘Never Trump‘ Republicans to agree with Trump,” he said. “The Republicans are going to drive home over and over that the Democrats are violating historic precedent, they are violating the president’s due process, and that the Democrats have pre-planned this from day one.”

⦁ David Sherfinski contributed to this report.

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