WASHINGTON — Mark Sanford, the former Republican governor of South Carolina who once brought squealing pigs into the State Capitol to make a point about pork barrel spending, is running against President Trump because of his failure to cut government waste.

“He’s not delivering on what he said he was going to do, which is eliminate the debt,” Mr. Sanford, who is also a former congressman, said in an interview. “In fact, it has gone in the opposite direction.”

Joe Walsh, the former Tea Party congressman from Illinois and a onetime Trump supporter who says he has now seen the light, is challenging the president as a moral mission. “I think he’s unfit and a danger to the country,” he said in an interview. “I do believe that most Republicans privately feel that way. This campaign is going to try and get them to say it publicly.”

And William F. Weld, the former Massachusetts governor, is staking his long-shot presidential hopes on appealing to moderates in New Hampshire, for whom supporting a pro-abortion rights, pro-legalized marijuana libertarian could be a meaningful protest vote against Mr. Trump.

Supporters of Mr. Trump’s Republican challengers refer to them as the “Three Musketeers,” and argue that having a trio of challengers — however long their long-shot bids are — could add up to enough of a nuisance to whittle away support for a vulnerable incumbent.

The president, on Twitter, has given them a more demeaning nickname: “the Three Stooges.”

The reality of their shared project of depriving Mr. Trump of his party’s nomination may fall somewhere between fearless and farcical. His challengers were all defeated in their last races and have little to lose in taking on what appears to be a fool’s errand: challenging a president whose approval rating in his own party ranks consistently in the high 80s.

But with three Republicans running — each representing a different constituency in the party — coupled with a softening economy and Mr. Trump’s own falling poll numbers against almost any Democrat, the theory of the case is that reluctant Trump voters may start to see a way out.

“The thing about Trump that’s so central to his power is that he beat Hillary Clinton when he wasn’t supposed to,” said Sarah Longwell, a Republican strategist who has been working to resist Mr. Trump from within the party. “He ran an inside straight. People have decided forget what people tell you about polls, and think this guy’s got magic.”

While many Republican lawmakers, lobbyists and operatives are still as privately contemptuous of Mr. Trump as they were when he effectively staged a hostile takeover of the party in 2016, they mute their criticism today. They are silent because they want to be re-elected, because they want to retain clients and access or because they are scared of making their states targets of Mr. Trump’s anger if they dare to speak out against him.

Even some anti-Trump Republicans now comfortably ensconced in retirement doubt he can be defeated as long as the party base remains so enamored of him.

“He could get down to the 20s in his overall approval and still have that subset of a subset that he needs for the primary,” said Jeff Flake, a former Arizona senator and a frequent target of the president. Mr. Flake said he did not seriously consider a primary race because he would rather “wait for this fever to cool” in his party.

And the Republican who could pose the biggest threat to Mr. Trump because of his donor relationships and name identification has said he is not interested.

In an email, Senator Mitt Romney of Utah wrote that he was “100 percent not running” and was likely to sit out the election. “Probably will not endorse any candidate for president,” he said.

Part of the challenge for any challenger is that Mr. Trump’s campaign operatives have worked for over a year to lock up support at the state level. They went out of their way to reward one loyalist, Jeff Kaufmann, the chairman of the Iowa Republican Party, with a fund-raiser headlined by Mr. Trump himself, hoping that other party leaders would take notice.

The Republican Parties in Arizona, Kansas, Nevada and South Carolina, meanwhile, are planning to cancel the 2020 presidential primaries in their states to make it virtually impossible for Mr. Trump’s challengers to build support.

The Republican National Committee, which is working hand in hand with Mr. Trump’s campaign, controls the debates, forcing two of the challengers to debate themselves on Tuesday at a forum hosted by the website Business Insider. (Mr. Sanford has said he will not participate.) Some of the candidates said they had had trouble getting booked with any regularity on Fox News.

“Fox has pretty much given me the back of their hand,” said Mr. Walsh, once a frequent guest.

Still, despite the hurdles — and the lack of any clear path to victory — the three men believe they can at the very least winnow Mr. Trump’s support, especially in New Hampshire, which holds the first nationwide primary. And they seem to be helping each other along the way.

“The Weld campaign has seen an uptick in interest and engagement since Congressmen Walsh and Sanford got into the race,” said R. J. Lyman, the chairman of Mr. Weld’s campaign. “Online fund-raising yields more on a daily basis than it did before.” Mr. Weld’s average donation has jumped to $50 from $25 since Mr. Walsh and Mr. Sanford entered the race, he said.

“It’s created this narrative that there are challengers to Trump,” Mr. Walsh said. “States canceling their primaries sure looks a hell of a lot worse with three challengers.”

And the challengers are working together to fight the canceling of primaries. Last week, they wrote an op-ed in The Washington Post, calling the move a “critical mistake.”

Mr. Walsh said the silence from Republicans had been disappointing. “We hear we’re not ‘Tier 1’ candidates, O.K., fine,” he said. “Where the hell are you, ‘Tier 1’ candidates? They want Trump to lose in 2020, they all believe he’ll lose and they’ve made a short-term bet that we can go back to the way we were before. I have very little patience for that — the time to stand up against Trump is now.”

In the past, when sitting presidents face heated primaries, they often failed to survive the general election. President Lyndon B. Johnson was effectively driven from seeking re-election after Eugene McCarthy showed surprising strength in New Hampshire in 1968. President Jimmy Carter had to contend with a serious challenge from Edward M. Kennedy for much of 1980, and lost re-election, as did President George Bush in 1992 after he was forced to contend with Patrick J. Buchanan.

Of the three challengers, Mr. Sanford could prove the most formidable when it comes to hectoring Mr. Trump from the right. He has argued that the devotion to Mr. Trump in the Republican rank-and-file could diminish if the economy slips.

“The moment that there is deterioration in the economy the value proposition of a Trump presidency goes out the window,” he said.

Mr. Trump has long detested Mr. Sanford, according to congressional aides. When Mr. Trump spoke at a meeting of House Republicans after the 2018 midterm elections, he was greeted with awkward silence and some boos when he singled out Mr. Sanford after his primary loss and said he “wanted to congratulate Mark Sanford on his big win.”

Mr. Sanford, however, is the only one of the three challengers who has not ruled out supporting Mr. Trump in a general election. Mr. Weld and Mr. Walsh, by contrast, said under no circumstances, now or in the future, would they ever get behind him.

For now, the campaigns are trying to gain legitimacy by bringing on serious names in Republican politics, even if they cannot get any real endorsements from lawmakers who are privately disdainful of Mr. Trump. Mr. Walsh recently hired Ann Herberger, a longtime fund-raiser for the Bush family, to oversee his fund-raising. Mr. Weld, meanwhile, announced a New Hampshire steering committee made up of people who have worked in Republican politics in the state for decades.

The Trump campaign, for its part, has publicly reacted to the growing field of challengers with a yawn and an eyeroll, even as it has worked with the party to shut down primaries.

“I have not had one conversation of even small concern over these three,” said Mr. Kaufmann, the Iowa Republican Party chairman. “Not even a hint and I’m talking about five to 20 text emails or phone conversations a day with the R.N.C. and the Trump campaign.”

Some Trump critics are hoping any attention the three challengers receive will prompt more to join them. “They are talented and will weaken President Trump and I am hopeful there will be more that enter the race,” said Anthony Scaramucci, the former White House communications director who has recently rebranded himself as a high-profile critic of Mr. Trump’s.

Still, the expectations are relatively low, even among the candidates themselves. Asked if he believed he could actually deny Mr. Trump the nomination next year, Mr. Sanford stopped short.

“I don’t get ahead of my skis on that front,” he said. “One step at a time.”

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