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Prominent Democrats, including some women thought to be on Joe Biden’s running-mate shortlist, have lined up behind the former vice president in the month since Tara Reade, a former Senate staffer, made public an allegation he sexually assaulted her once in the 1990s.

Sens. Amy Klobuchar of Minnesota and Kamala Harris of California and former Georgia House Minority Leader Stacey Abrams all have said that they respect women making such allegations and take them seriously. But they haven’t wavered in their support for Mr. Biden, the presumptive Democratic presidential nominee, whose campaign has strongly denied the allegations.

For the most part, Democrats haven’t voiced their opinions on the accuracy of Ms. Reade’s claims. Instead, they have attested to the longtime lawmaker’s character and policy efforts. California Rep. Ro Khanna, a co-chairman of Vermont Sen. Bernie Sanders’s presidential campaign, said President Obama had assessed his former No. 2, and “for me, that’s the ultimate vetting.” Mr. Sanders endorsed Mr. Biden on April 13, after suspending his campaign a week earlier.

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For her part, Ms Reade said she has been deflated by the lack of reaction over the past month, feeling that she has little support even from #MeToo advocates. Still, she says, she has no regrets about telling her story.

“I can’t get restorative justice, but I wanted to further the sexual-harassment conversation, particularly about how powerful men can use the system,” she said.

Mr. Biden, a Delaware senator from 1973 to 2009, hasn’t commented on Ms. Reade’s allegations. His campaign and former aides who worked in his office at the time have denied that any woman, including Ms. Reade, was mistreated by Mr. Biden or brought any concerns to them. In the last month, he has participated in about 20 television interviews as well as podcasts and virtual fundraisers. He hasn’t been asked about Ms. Reade’s claims at any of those events.

Ms. Reade, 56 years old, has made two sets of allegations against Mr. Biden, 77, both of which stem from her time working in his Senate office for less than a year beginning in December 1992, when she was in her late 20s. She has said for more than a year that while she worked for him, Mr. Biden repeatedly rubbed her shoulders and neck in ways that made her uncomfortable.

Late last month, Ms. Reade made a new allegation public. She said in a podcast interview and subsequent interviews with media including The Wall Street Journal that Mr. Biden had once cornered her in a corridor of the U.S. Capitol complex and touched her genitals without permission.

Ms. Reade said she told her mother, brother and several friends about her alleged experience at the time it happened. She also said she reported the alleged touching that made her uncomfortable, but not the alleged assault, to three high-level Biden staff members.

Ms. Reade also said she filed a complaint with the Senate’s personnel office about the alleged sexual harassment—though not the alleged assault—but nothing happened. She said she was forced out of the job as Mr. Biden’s coordinator of interns after making the reports. She said she doesn’t have a copy of her complaint to the office, and efforts to locate it with the Senate were unsuccessful. The office no longer exists.

The Biden campaign has denied all of Ms. Reade’s allegations. Kate Bedingfield, the campaign’s communications director, said in a statement: “Vice President Biden has dedicated his public life to changing the culture and the laws around violence against women. He authored and fought for the passage and reauthorization of the landmark Violence Against Women Act. He firmly believes that women have a right to be heard—and heard respectfully. Such claims should also be diligently reviewed by an independent press. What is clear about this claim: it is untrue. This absolutely did not happen.”

Mr. Biden’s campaign declined to make him available for an interview for this article.

Ms. Reade’s brother and a friend who had been a Capitol Hill intern in other offices both told the Journal that Ms. Reade had told them about the alleged assault, which she said occurred after her supervisor had asked her to take a duffel bag to Mr. Biden. She said he kissed her and reached under her skirt to put his fingers inside her vagina.

Ms. Reade, her brother and her friend couldn’t remember the exact date of the alleged assault. Ms. Reade’s friend said she recalled discussing it on two phone calls with Ms. Reade the day after Ms. Reade said the assault occurred; the friend said she took the calls in her college dormitory room and remembered it being late in the spring semester of 1993.

Ms. Reade also said she had long, tearful conversations with her mother, Jeanette Altimus, about the alleged incidents throughout her time working for Mr. Biden. She said her mother encouraged her to quit and, after the assault allegedly occurred, to file a police report. Ms. Reade said she didn’t take the advice.

Ms. Altimus died in December 2016, but Ms. Reade said her mother had once discussed her situation in vague terms, without naming Mr. Biden, during a call to CNN’s “Larry King Live.”

The call was anonymous, and Ms. Reade couldn’t recall when the episode aired. During an Aug. 11, 1993, show, a caller from San Luis Obispo, Calif., said her daughter had left “after working for a prominent senator, and could not get through with her problems at all.”

After CNN recently aired the clip, Ms. Reade said she is certain the caller was her mother, who lived in San Luis Obispo at the time.

The three former Biden aides whom Ms. Reade said she told about the alleged touching—longtime Biden executive assistant Marianne Baker, chief of staff Ted Kaufman and policy aide Dennis Toner—all denied she ever told them anything like that.

In a statement provided by the Biden campaign, Ms. Baker said she had worked for him for 20 years and “never once witnessed, or heard of, or received, any reports of inappropriate conduct, period—not from Ms. Reade, not from anyone. I have absolutely no knowledge or memory of Ms. Reade’s accounting of events, which would have left a searing impression on me as a woman professional, and as a manager. These clearly false allegations are in complete contradiction to both the inner workings of our Senate office and to the man I know and worked so closely with for almost two decades."

Messrs. Kaufman and Toner both told the Journal they didn’t remember Ms. Reade.

“I don’t remember, and I would have remembered her if she’d come to me,” said Mr. Kaufman, who has known Mr. Biden for almost five decades.

Mr. Toner, who worked for Mr. Biden for 34 years, also said he would have recalled a conversation of that nature.

“I do not remember any such conversation that I was alleged to have with Tara regarding Joe Biden and alleged sexual behavior,” he said. “I would recall it, I would remember the date. But it didn’t happen. That’s not Joe Biden.”

As a lifelong Democrat, Ms. Reade said in an interview, she “always felt conflicted about Biden,” whose legislative efforts, such as sponsoring the Violence Against Women Act, she admired. Ms. Reade supported Mr. Sanders during the primary, but she posted social-media messages supportive of Mr. Biden as recently as 2017 and said she voted twice for Mr. Obama in presidential elections.

She said she decided to come forward with the allegation of inappropriate touching after reading about other women who said Mr. Biden had touched them in ways that made them uncomfortable. Mr. Biden said in April 2019 of those complaints: “I hear what they’re saying. I understand it, and I’ll be much more mindful. That’s my responsibility.”

Ms. Reade said she was prepared to tell reporters about the alleged assault last year, too, “but I lost my courage.” She said she decided to speak up when Mr. Biden became the likely Democratic nominee.

Ms. Reade has previously given other explanations for leaving Mr. Biden’s office. In a 2018 Medium post, said she left because of what she considered Washington’s xenophobic attitude toward Russia and so that she could pursue an acting and artistic career.

Ms. Reade said she recently contacted the District of Columbia Metropolitan Police. The statute of limitations had passed, but Ms. Reade said she believed filing a police report would possibly encourage other women to come forward.

The public version of the incident report doesn’t contain names but says the filer “disclosed that she was the victim of a sexual assault which was committed by Subject-2 in 1993.” Ms. Reade said she named Mr. Biden in her police interview.

A police spokesman described the investigation as “ongoing” and declined to answer further questions.

President Trump’s allies have seized on the allegations of Ms. Reade and other women who say they had uncomfortable interactions with Mr. Biden. Mr. Trump’s son, Donald Trump Jr., is among those who have promoted Ms. Reade’s claims on Twitter, calling Mr. Biden “creepy.” Numerous women have accused Mr. Trump of sexual misconduct over the years. Mr. Trump has denied those allegations.

Aides to Mses. Klobuchar, Harris and Abrams and to Michigan Gov. Gretchen Whitmer pointed the Journal to comments they had made in previous interviews and declined to comment further.

“Women have the right to be heard, and we have the responsibility to listen,” Ms. Abrams told the Daily Beast. She said that Mr. Biden “has spent over 40 years in public life advocating for women,” and that nothing in an April 12 New York Times article about Ms. Reade’s allegations changed Ms. Abrams’s opinion of him.

Ms. Whitmer told HuffPost she “believes that it is important that these allegations are vetted, from the media to beyond, and that it is something that no one takes lightly.” She added: “But it is also something that is personal. We will not speculate or provide greater insight, without knowing more about the situation.”

Ms. Klobuchar told National Public Radio that all women have a right to be heard and for their claims to be reviewed. Of Mr. Biden, she said he is “someone I worked with, I see him on, a leader on, domestic abuse."

In a podcast interview with the San Francisco Chronicle, Ms. Harris had a similar sentiment.

She said Ms. Reade “has a right to tell her story. And I believe that, and I believe Joe Biden believes that, too.”

However, she said, the Mr. Biden she knew has "been a lifelong fighter, in terms of stopping violence against women.”

Write to Julie Bykowicz at julie.bykowicz@wsj.com