The movement is called “Bernie or Bust,” and it means just that: If Sen. Bernard Sanders of Vermont loses the Democratic presidential nomination, a group of his supporters will either write in his name in the general election or consider casting their ballot for a Republican.

The one thing they certainly won’t do: Vote for Hillary Clinton.

More than 50,000 people already have signed up at the Revolt Against Plutocracy, pledging to vote for the Green Party candidate in the general election or write in Mr. Sanders’ name if Mrs. Clinton wins the Democratic nomination. Other groups, such as Grassroots Action for Bernie, are taking to social media, using Facebook and Twitter to try to get the “Bernieorbust” hashtag trending.

Even Sanders supporters not tied to the movement, or unaware of its existence, seem to agree with its principles, making one thing clear: The Democratic National Committee and Mrs. Clinton will have a hard time attracting many of Mr. Sanders’ voters.

“I will not be voting for Clinton if Sanders does not win the nomination,” said Jon Clemens, a Sanders supporter from Hartford, Connecticut. “She has done nothing to earn my vote, and the Democratic Party should not assume that she will simply absorb Sanders’ supporters. Clinton has only ‘evolved’ to progressive political stances when public polling indicated to her that it was politically advantageous to do so. She is disingenuous, has little integrity and lacks vision.”

As Mrs. Clinton gets closer to sewing up the nomination, her campaign will begin to grapple with damage from an increasingly divisive fight with Mr. Sanders.

Mrs. Clinton spent much of the primary contest tacking to the left to try to blunt Mr. Sanders’ attacks, but the senator’s supporters say her late-season political conversion isn’t convincing.

“We Bernie fans just won’t vote for her,” said Steph Faulkner, who hails from Mr. Sanders’ home state of Vermont and is an avid Sanders supporter. “We are sick of the media telling us we have too. We don’t like her. We don’t trust her. We believe she is a Wall Street puppet. There is nothing they can say that will make us vote for such a woman. I mean, heck, people would vote for Trump over her, and he is a monster. What does that tell you? It tells me she is seen as the bigger evil. Trump is less evil than Hillary.”

Part of the equation for Mrs. Clinton and other Democratic leaders is how widespread that sentiment is within their political base. Sanders supporters say not to underestimate them.

“More than 50 percent of Sanders supporters will never vote for her,” Chris Fox, a Sanders supporter in Fairfield, Ohio, said in an email. “That is why she will not beat Trump. Weigh the Republican hate for her (motivation to vote) against the Democrat progressive liberal’s hate for the status quo (unmotivated to vote for her), and we have a major problem on our hands. Only Sanders can beat Trump, but that’s not why we are voting for him. He’s the only person we trust with the job.”

The sentiment was by no means universal. At the polls Tuesday, a number of Sanders supporters told The Washington Times that despite tension between Mrs. Clinton and Mr. Sanders, they would vote for the former secretary of state if only to stop Republicans.

“Any of the Republicans would move us more toward the killing of civilians and a more interventionist foreign policy than what we’ve seen from the Obama administration,” said a 30-year-old woman voting in Atlanta.

For some Sanders voters, their decision in November will depend on Republicans. The prospect of businessman Donald Trump winning the Republican nomination and facing off against Mrs. Clinton would force many of the Sanders supporters into Mrs. Clinton’s camp.

Steve Herbert, a technology consultant voting in Atlanta, who called Mrs. Clinton “wishy-washy” on major political issues, said he would have to back her in a race against Mr. Trump, but it would be a tossup if she faced any other Republican.

“It depends on what comes out on the other side,” Mr. Herbert said.

Mrs. Clinton’s backers said they are not afraid of massive defections and are counting on the anti-Trump vote to bring Sanders supporters back into the Democratic fold.

“Right now, emotions are really hot. I know what mine were when Barack Obama won,” said Maureen Rehg, 60, a Clinton volunteer. “So I know how they feel to really be passionate about someone, and I think Bernie Sanders has a lot of good ideas — a lot of good ideas — and he is a good man, but I think she is more qualified.”

Several voters said the looming Supreme Court fight could chase voters to Mrs. Clinton’s corner, with Democrats fearful of a Republican nominating the next justice.

Mr. Fox, the Sanders supporter in Ohio, said he might be forced into a Clinton vote if the court nomination isn’t settled. But he said if a replacement for the late Justice Antonin Scalia is in place, he would write in Mr. Sanders’ name on his November ballot.

Other Sanders supporters say a Clinton nomination would hand the election to Mr. Trump — and some may even vote for him, enticed by his vow to finance his own campaign and his promise to remain outside the control of special interests.

“I like the nonestablishment side — people not owned by big banks or businesses to do their bidding. We need to change how Washington works,” said Jadon Salvant, a Sanders supporter from Fairfax, Virginia.

Distrust of establishment politics runs deep among Sanders supporters, and that particularly dents Mrs. Clinton, who has been a first lady for eight years, a senator for eight years and a secretary of state for four years.

Her ties to Wall Street, her use of a secret email server while head of the State Department and her unwillingness to release transcripts of speeches she made to Wall Street executives feed the anxiety.

Sanders supporters also say they will blame the Democratic National Committee if their candidate loses the nomination. They point to DNC Chairwoman Debbie Wasserman Schultz’s early decision to limit the number of debates, her support in 2008 for Mrs. Clinton’s bid and her feud with Mr. Sanders over the DNC’s voter files, as evidence that she was orchestrating a Clinton nomination from the start.

Mr. Sanders’ fans also blame the media, which they say has been unfair in its coverage of him, despite the crowds he has drawn at his rallies. According to a Decisiondata.org study completed in January, Mr. Sanders received 29,525 mainstream media mentions from June to January, compared with Mrs. Clinton’s 87,737 and Republican front-runner Donald Trump’s 183,903.

“It’s really sad how the press has tried to silence Bernie,” said Mark Hartung, a Sanders supporter. He said even liberal-leaning MSNBC shifted its coverage away from progressive programming and toward establishment Democratic views. He blamed corporate interests he said were trying to silence progressive voices.

“I think I’ll just write Bernie in and I think the DNC should look over their shoulders because all they want to do is keep the money rolling,” he said. “The Democrats have tried to hand this primary to Hillary Clinton and bypass the will of the American people, and it shows just how much money is influencing the process. Tell them to be scared because the young crowd is coming for them. They are finished.”

• S.A. Miller, reporting in Atlanta, and Seth McLaughlin, reporting in Virginia, contributed to this report.

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