WASHINGTON — He would relish a run against Senator Elizabeth Warren and has dismissed Senator Bernie Sanders as “crazy.” He has worried about Joseph R. Biden Jr.’s standing in the polls, at the same time telling advisers that the former vice president is a male version of Hillary Clinton and that a matchup in 2020 is a race he knows how to run.
But President Trump has been less sure-footed about how he would take on Senator Kamala Harris if she were to be his rival. And after her breakout performance in the Democratic presidential primary debate on Thursday night, Trump campaign aides and allies acknowledged that Ms. Harris, a black former prosecutor from the donor-laden state of California, could prove to be a vexing adversary.
“I would be more worried about Kamala Harris than Joe Biden as the nominee,” Matt Schlapp, the chairman of the American Conservative Union, said after watching the debate. “The Joe Biden of last night is a candidate I would dream to run against. But she can put together the Obama coalition. She’s also a first, and Democrats like the tingles up the leg.”
In one lacerating rebuke of Mr. Biden on Thursday night, a man two decades her senior who has held some of the most powerful posts in American government, Ms. Harris altered two perceptions about the 2020 election. She gave an audience of millions a preview of how she could be a nimble opponent in a televised face-off with Mr. Trump. And she chipped away at some of Mr. Biden’s aura of dominance in the Democratic field.
Many Republicans said even before Thursday’s debate in Miami that Mr. Biden’s presumed strengths were overstated — a sentiment the president shares. They described him as “old” and “tired” as well as someone they believed Mr. Trump knew how to spar with and rattle in a head-to-head contest.
“If this is a window into how Biden would perform against Trump, it cannot give Democrats confidence,” said Ralph Reed, the founder of the Faith and Freedom Coalition, who is in frequent contact with the White House. Mr. Reed’s organization is one of a host of social conservative groups that will be spending tens of millions of dollars to identify and turn out voters for Mr. Trump in 2020.
At a minimum, what some Republicans assumed would be a straightforward contest with Mr. Biden has suddenly become more unpredictable because of the Democratic debates.
Ronna McDaniel, the chairwoman of the Republican National Committee, said in an interview that Mr. Biden had fallen “flat on his face.” That, she said, pointed to a more protracted and contentious battle for the Democratic nomination than many Democrats — and Republicans — had hoped for.
“You’re not going to have one set front-runner throughout,” Ms. McDaniel said. “There were two schools of thought. Is Biden going to carry the lead all the way to the end? Or are we going to have a lot of competitive candidates fighting this out?”
What became clear, she said, was that “there’s the potential for many different candidates to take the lead.”
Privately, people close to the White House said they viewed Ms. Harris as “very dangerous” and the hands-down winner of the debate on Thursday. Others said they hoped the debate would serve as a warning for Republicans — including those in the president’s inner circle — who have been too dismissive of Mr. Biden’s less seasoned rivals.
Still, they said the debate did not change the fundamental assumption that campaign officials and allies of the president’s have made about the dynamics of the race. And they cautioned that underestimating Mr. Biden’s strength as a formidable force in the Democratic primary and as a serious threat to Mr. Trump in the general election would be a mistake.
Until now, the Trump campaign and Republican Party strategists have lumped in Ms. Harris with the rest of the Democratic field, planning to pigeonhole her as a far-left radical in the mold of Mr. Sanders. They noted that like Ms. Warren and Mr. Sanders, she raised her hand in the debate to signal support for abolishing private health insurance — an issue the Trump campaign expects to be the easiest way to explain what the Democratic version of “socialism” looks like.
(On Friday morning, Ms. Harris walked back her position, saying she misunderstood the question and does not support abolishing private health insurance.)
Trump aides and allies were equally pleased when she put Mr. Biden on his back foot, confronting him over his defense of Southern senators and support of antibusing policies decades ago. But even as they took pleasure in Mr. Biden’s bad night, they conceded that her strong performance made it clear that Ms. Harris, too, could pose a real threat.
Ms. Harris, who has demonstrated the natural political skills closer to those of President Barack Obama than those of Mrs. Clinton, would present Mr. Trump with an unfamiliar rival, and one whose race and gender are potential land mines for him in terms of his attacks. So far, he has struggled to land on a quick-hit way to undermine Ms. Harris’s candidacy. Aides said that was because her poll numbers had not been high enough to register with him.
After Ms. Harris’s kickoff rally in January, he complimented her crowd size, describing it as “the best opening so far.” In an interview with the Fox News host Sean Hannity, the president said that “she’s got a little bit of a nasty wit.” But Mr. Trump has focused most of his energy on Mr. Biden, whom internal polls have shown beating him in critical states like Pennsylvania, Michigan and Ohio.
Some Republicans would welcome a Trump-Harris race because it would allow them to frame the election exactly as they hope to: a battle of ideologies that will excite their conservative base with dire warnings of a socialist future.
“This is a bit of a ‘Braveheart’ election, where everybody’s painted their faces blue and they’re all running at each other, and there’s no quarter taken or given,” Mr. Reed said. “The center is hollowed out. The unaffiliated voter has not disappeared but they’re statistically insubstantial. So what you’re looking for in that kind of election is a thing that increases intensity, enthusiasm and a feeling that something is really on the line.”
The effort to paint the Democratic Party in a radical light is expected to be well coordinated and led by a coalition of some of the biggest groups in Republican politics. “This lurch to the left and full throttled assault on the free enterprise system will be wrapped around the neck of the eventual Democratic nominee by the G.O.P.,” said Scott W. Reed, the chief political strategist for the U.S. Chamber of Commerce.
“No socialismo, no comunismo, somos capitalista,” read a large banner hanging between two palm trees at a demonstration of Trump supporters on Wednesday outside the debate hall in Miami.