LAS VEGAS — Pete Buttigieg liked the line so much, he used it twice.
Asked at back-to-back rallies on Monday what would happen if President Trump claimed election fraud and refused to vacate the White House for him, Mr. Buttigieg said, “If he really doesn’t want to leave, we could work something out, but he’d have to do his chores.”
After a burst of laughter, he said that Democrats couldn’t merely “eke out a win” in November; they must earn a large majority that includes taking the Senate. “So I’m determined to build the broadest, biggest coalition possible,” Mr. Buttigieg said in Carson City, Nev., repeating a message he has leaned on heavily before the Nevada caucuses on Saturday.
Yet Mr. Buttigieg’s efforts at coalition-building — a strategy that helped deliver him top-two finishes in Iowa and New Hampshire — are at risk of falling short now that the primary has come to racially diverse states. The difficulty that Mr. Buttigieg has long had winning over people of color and younger voters has become a more obvious liability. As Senator Bernie Sanders of Vermont builds momentum and the former New York City Mayor Michael R. Bloomberg emerges as a contender for the anti-Sanders vote, Mr. Buttigieg’s path to the Democratic nomination may soon be narrowing.
For months, as Mr. Buttigieg poured time and money into Iowa, he told skeptics that winning would beget winning. Even though the first two states are nearly all white, he said that top finishes there would bring more black and Latino supporters.
There is not much sign of that so far in Nevada or South Carolina, which votes Feb. 29.
Mr. Buttigieg, the former mayor of South Bend, Ind., said in an interview that he expected his effort in Nevada — more than 100 organizers, a packed events schedule, an ad that he narrates in Spanish — would pay off. “We’re putting in the work,” he said. “You’ve got to earn it.”
“One thing we’re also seeing is a lot of voters will be making a final decision very late in the game,” he added. “Every day of distance run here is going to pay off.”
Undecided voters who have come to hear Mr. Buttigieg in the days before Saturday’s caucuses — as well as in Sacramento, Calif., on Friday — often left as converts.
But Mr. Buttigieg may simply be running out of runway for his candidacy as Nevadans completed four days of early voting on Tuesday, and delegate selection is set to move into high gear in the 14 Super Tuesday states.
Mr. Bloomberg’s addition to a nationally televised debate on Wednesday in Las Vegas may also draw an unflattering contrast between the two former mayors: one who oversaw a city of 100,000, and one who led a metropolis of eight million. Mr. Buttigieg argued that Mr. Bloomberg has gained a spot on the debate stage thanks to lavish spending, but that Democrats should beware.
“There’s more to winning an election than how much money you put in,” he said. “After all, President Trump, I believe, was outspent in the last general election. That should be a warning to anybody who thinks that money alone is going to make somebody the right candidate to defeat Donald Trump.”
On Tuesday, a Monmouth University poll of Virginia, a diverse state that votes on Super Tuesday, showed Mr. Buttigieg barely in double digits, trailing three others.
Nevada polls, while scarce, suggest it was Mr. Sanders who gained the most ground after winning New Hampshire by an unexpectedly close margin over Mr. Buttigieg.
A Telemundo poll of likely Latino caucusgoers in Nevada that was released Tuesday showed former Vice President Joseph R. Biden Jr. with 34 percent support, Mr. Sanders with 31 percent and Mr. Buttigieg a distant third with 7 percent.
When asked about immigration, as he often is here in a state with a population that is 30 percent Latino, Mr. Buttigieg calls for a path to citizenship for undocumented people and speaks about how Hispanic newcomers to South Bend reversed a decades-long population decline and brought economic vigor. At a rally in Las Vegas on Sunday morning, he answered in Spanish at length to a Spanish-speaker’s question, which led to a chant of “Si se puede.”
That evening in Sparks, Nev., after hearing him at a high school gym, Senen Bonilla, a Mexican immigrant who became a citizen four years ago and plans to caucus for the first time, said she had leaned toward Mr. Sanders but decided that day on Mr. Buttigieg. “He’s young and has a lot more energy,” she said.
Paul Traudt, a Buttigieg supporter and donor in Las Vegas, said: “I’m realistic. I’m not optimistic.” He predicted a third- or fourth-place finish in Nevada for Mr. Buttigieg.
Mr. Traudt, a retired media studies professor, said that as a donor to the campaign, he was getting more incessant appeals to contribute, which might reflect that Mr. Buttigieg spent not just lots of his time but also much of his treasure in Iowa and New Hampshire.
“I’m getting texts from two or three different parties, and it’s just this urgency and ‘we need more money, we need more money, we need more money,’” Mr. Traudt said.
In addition to being nearly 30 percent Latino, Nevada is about 10 percent black and 10 percent Asian-American. Dina Neal, a member of the State Assembly whose district in North Las Vegas is 25 percent black, said that despite Mr. Buttigieg’s robust organizing, he had not made inroads with black caucusgoers.
“There are some people who feel strongly” that Mr. Buttigieg has “not properly courted the African-American community,” said Ms. Neal, who endorsed Mr. Biden.
She attended a brunch on Sunday hosted by the Black Legislative Caucus, where Mr. Buttigieg told the room about his Douglass Plan to end institutional racism. He seemed to draw little attention from the 200 guests, who chatted and served themselves at a buffet.
Attendees mentioned Mr. Buttigieg’s lack of experience as the 38-year-old former mayor of a small city, as well as news accounts of opposition he faced by some black residents of South Bend.
In Nevada, Mr. Buttigieg’s message and strategy are similar to what worked for him in Iowa: Organize and travel widely, especially in areas where Democrats don’t often compete. On Monday, he campaigned in Elko, a deeply Republican region. He drew 300 people, but the effort may pay small dividends: Only 88 Democrats caucused in 2016 in all of Elko County, one of Nevada’s largest by area.
Democratic voters are overwhelmingly concentrated in just two places: Clark County (Las Vegas) and Washoe County (Reno).
While Mr. Biden has attacked Mr. Sanders in recent days, and Mr. Sanders has rounded on Mr. Bloomberg, Mr. Buttigieg has stood largely apart from the fighting. On Tuesday night, his campaign announced that on Wednesday he would join members of Nevada’s powerful Culinary Workers Union, which declined to endorse a candidate, in a picket outside the Palms Casino.
Except for the faintest of swipes at Mr. Sanders in a TV ad here — saying Democrats can’t defeat Mr. Trump by “overreaching” — Mr. Buttigieg has portrayed himself as welcoming independents and “future former Republicans.”
“One of the things that’s really important to me in building this campaign is reaching out to everybody,” he said in Reno. “I’m never going to push somebody away who maybe doesn’t agree 100 percent of the time.”
And at his rallies, there are always Republicans who have soured on Mr. Trump and say they like Mr. Buttigieg’s promise to end the country’s bitter polarization.
“His tone, his believability factor, are just a lot stronger than most of the candidates,” said James Sohl, a Republican whose wife lured him away from work to Mr. Buttigieg’s rally in Carson City.
The night before, Bob Swiatek, a Republican who is a retired law enforcement officer, said he could vote for Mr. Buttigieg as the Democratic nominee.
“There’s a lot of Republicans out there that aren’t talking that are going to go the other way” in November, he said.
But running as a candidate who would appeal to independents and Republicans in the general election isn’t helping Mr. Buttigieg as he seeks a primary victory. Neither Mr. Sohl nor Mr. Swiatek planned to change their party registrations to caucus for Mr. Buttigieg.
With Mr. Buttigieg pinning his hopes on late-deciding voters, some who have come to hear him in the last days left persuaded by his message.
“I came not sure if I’d put him No. 1 or Amy No. 1,” Ron Procter said after a Buttigieg event for veterans in Reno, referring to Senator Amy Klobuchar of Minnesota. “He’s definitely No. 1.”
A Vietnam combat veteran, Mr. Procter said he was impressed by Mr. Buttigieg’s quick responses to audience questions, in which he talked about veteran suicides and immigration, as well as his motivation for joining the Navy Reserve.
“I was worried about him being on the stage with Trump, but I think this guy can handle him,” he said. “He seems to have a presence that I’ve seen on some really good leaders in my life. I don’t think he intimidates very easy.”