Fergus Cullen, a GOP fixture in New Hampshire politics, sent out 150 invites to his house party for Bill Weld, the first Republican to announce a primary challenge to President Trump.

No fan of the president, Mr. Cullen didn’t go out of his way to invite die-hard Trump supporters, and some simply couldn’t make it. But more than 60 people showed up and — most notably — only three declined with some version of “Thanks, but I’m with Trump.”

“I thought I’d get more,” Mr. Cullen said of Trump die-hards. “It confirmed for me that people are open to an alternative. Doesn’t mean they’ll all vote for an alternative, but they are open to hearing from someone else.”

New Hampshire is critical ground for the “Never Trumpers” who hope to defy the odds and mount an intraparty challenge to a sitting president.

Mr. Weld, a former Massachusetts governor, announced an exploratory committee in February and made it official this month by saying he is challenging Mr. Trump.

Maryland Gov. Larry Hogan also is eyeing the race. He stopped at a Politics and Eggs event in Manchester, New Hampshire, last week as part of a 16-state listening tour.

Ohio Gov. John Kasich, who has flirted with taking on Mr. Trump again after losing to him in 2016, is waiting in the wings. He said both parties are failing to tackle issues such as climate change and immigration.

“There is a market for an anti-Trump,” said Tom Rath, a former New Hampshire attorney general and key Republican Party figure in the state. “But I don’t know how big it is. I don’t know how active it’s going to be.”

Independents make up 40% of the Granite State electorate and can choose either party primary ballot. Those who don’t find a champion on the Democratic side next year could play a role in the Republican contest.

“New Hampshire rarely likes to do the expected thing. It’s a state that takes its responsibility very seriously and really studies the candidates,” said Stuart Stevens, a Republican Party consultant who is advising Mr. Weld. “There’s a history of insurgent campaigns doing well there, and I think Bill Weld will do very well in New Hampshire.”

Mr. Weld says he is running because America “deserves better” than Mr. Trump. He cited the president’s belligerent manner, his attitude toward Russia and other eyebrow-raising moments, such as his comments after the riots in Charlottesville, Virginia.

Mr. Hogan is more circumspect.

Speaking at St. Anselm College, he said he is not in a rush to beat New Hampshire’s filing deadline in November and that he has no interest in launching a “suicide mission” while he holds his day job as governor.

Mr. Hogan said he has heard from people who want him to run, yet he focused on his bipartisan record in Maryland during his Manchester stop instead of rolling out a stump speech.

“He got good reviews internally, in terms of the chattering class, but it’s not clear yet what he’s doing,” Mr. Rath said.

Any challenge to Mr. Trump is a long shot. National polls suggest the president has support from 90% of the party.

A University of New Hampshire poll in February found two-thirds of likely Republican primary voters in the state would support Mr. Trump and about one-sixth would back Mr. Kasich. Mr. Weld drew just 3%.

The poll was taken before the release of the special counsel’s report finding no conspiracy to subvert the 2016 election but suggesting that Mr. Trump’s attempts to thwart investigations into Russian meddling might be considered obstruction of justice.

Mr. Hogan said the report detailed “disturbing” behavior. Mr. Weld dubbed Mr. Trump a “one-man crime wave.”

Their concerns run deeper than Mr. Mueller’s 448 pages, however. They argue that the party is losing its grip on values such as fiscal and personal responsibility, a hard line on Russia and making sure “character counts.”

“On every one of those issues, Donald Trump is on the other side,” Mr. Stevens said. “That’s true if there’d never been a Mueller report, it’s true with the Mueller report.”

Former Sen. Bob Corker, a Tennessee Republican who tangled with Mr. Trump before retiring this year, said a primary would help Republicans focus on bread-and-butter issues such as responsible budgeting.

“Philosophically, you could look at it and say that it would be a good thing for our country should that occur from the standpoint of issues,” Mr. Corker told the Time 100 Summit in New York City. “If you had a real primary, where you had someone that was really being listened to, and of substance, things that we were talking about — and I could go through a list of them — they would actually be debated in a real way.”

It has been a while since a sitting president faced a stiff challenge from within his own party, though it’s hardly unprecedented.

Republican Pat Buchanan challenged President George H.W. Bush in 1992 on a platform that presaged some of the populist themes Mr. Trump rode to the presidency. Although he captured 37% of the New Hampshire vote, he failed to win any state primaries.

Sen. Edward M. Kennedy of Massachusetts managed to win a few states in his Democratic primary challenge against President Jimmy Carter in 1980. The peanut farmer from Georgia ultimately staved off Kennedy’s challenge before losing in a landslide to Republican Ronald Reagan, who had challenged President Ford in the 1976 Republican race.

Yet a sitting president has not been denied a nomination that he wants since the 19th-century, smoke-filled rooms that preceded the modern primary system.

“I will be surprised if anybody besides Trump gets anywhere,” said Doug Wead, a conservative commentator and presidential historian. “In my humble opinion, the media is so suffocating and strident and they can’t help themselves. But it drives voters to do the opposite. So they will be promoting Weld or anybody else, and that will rebound to Trump.”

The Republican National Committee says there is no reason for a divisive fight.

“President Trump received more primary votes than any nominee in Republican Party history, and he now enjoys the highest approval rating in his party of almost any president in modern history,” RNC spokesman Steve Guest said. “The RNC and the Republican Party are firmly behind the president. Any effort to challenge the president’s nomination is bound to go absolutely nowhere.”

Mr. Cullen, though, said Mr. Trump should be worried by the rumblings because every sitting president who has faced a significant challenge in New Hampshire’s primary since World War II has failed to win the general election.

In addition to Ford, Mr. Carter and Bush all losing in November, two postwar presidents decided not to seek re-election after a humiliation in New Hampshire. President Harry S Truman decided to retire after his 1952 loss there, and President Lyndon B. Johnson left the race soon after his near loss in 1968.

“One candidate will end up serving as the vessel for those of us who want to express our opposition to and disgust with this administration,” Mr. Cullen said. “And there are many of us. Is it 10, 20 or 37%? Too early to tell.”

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