When Jimmy Kimmel asked Hillary Clinton in a late-night TV interviewabout U.F.O.s, she quickly corrected his terminology.

“You know, there’s a new name,” Mrs. Clinton said in the March appearance. “It’s unexplained aerial phenomenon,” she said. “U.A.P. That’s the latest nomenclature.”

Known for her grasp of policy, Mrs. Clinton has spoken at length in her presidential campaign on topics ranging from Alzheimer’s research to military tensions in the South China Sea. But it is her unusual knowledge about extraterrestrials that has struck a small but committed cohort of voters.

Mrs. Clinton has vowed that barring any threats to national security, she would open up government files on the subject, a shift from President Obama, who typically dismisses the topic as a joke. Her position has elated U.F.O. enthusiasts, who have declared Mrs. Clinton the first “E.T. candidate.”

“Hillary has embraced this issue with an absolutely unprecedented level of interest in American politics,” said Joseph G. Buchman, who has spent decades calling for more transparency in government about extraterrestrials.

Mrs. Clinton, a cautious candidate who often bemoans being the subject of Republican conspiracy theories, has shown surprising ease plunging into the discussion of the possibility of extraterrestrial beings.

She has said in recent interviews that as president she would release information about Area 51, the remote Air Force base in Nevada believed by some to be a secret hub where the government stores classified information about aliens and U.F.O.s.

In a radio interview last month, she said, “I want to open the files as much as we can.” Asked if she believed in U.F.O.s, Mrs. Clinton said, “I don’t know. I want to see what the information shows.” But, she added, “There’s enough stories out there that I don’t think everybody is just sitting in their kitchen making them up.”

When asked about extraterrestrials in an interview with The Conway Daily Sun in New Hampshire late last year, Mrs. Clinton promised to “get to the bottom of it.”

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“I think we may have been” visited already, she said in the interview. “We don’t know for sure.”

While Americans typically point to issues like the economy and terrorism as top priorities for the next president, a desire for answers about aliens has inspired a passionate bloc of voters, who make their voices heard on social media.

Stephen Bassett, who lobbies the government on extraterrestrial issues, views a Hillary Clinton presidency as a chance to finally get the United States to disclose all it knows about life beyond Earth. Since November 2014, Mr. Bassett’s organization has sent roughly 2.5 million Twitter messages to presidential candidates, elected officials and the news media urging a serious discussion of the issue. “That was a storm and now it’s like a steady drip,” Mr. Bassett said.

The movement viewed Mrs. Clinton’s decision to correct Mr. Kimmel’s use of the term U.F.O., which some view as loaded and more rooted in science fiction rather than in science, in particular, as a breakthrough because it “suggested she’d been briefed by someone and is not just being flippant,” Mr. Buchman said.

In fact, Mrs. Clinton had been briefed. She was prepped by her campaign chairman, John D. Podesta, who is not only a well-respected Washington hand, having served as a top adviser to Mr. Obama and President Bill Clinton, but is also a crusader for disclosure of government information on unexplained phenomena that could prove the existence of intelligent life outside Earth.

“The time to pull back the curtain on the topic is long overdue,” Mr. Podesta wrote in his foreword for the 2010 book “UFOs: Generals, Pilots and Government Officials Go on the Record” by Leslie Kean, an investigative journalist.

Mrs. Clinton’s position is not a political response to public sentiment — 63 percent of Americans do not believe in U.F.O.s, according to an Associated Press poll. But it reflects the decades of overlap between the rise to power of Bill and Hillary Clinton and popular culture’s obsession with the universe’s most mysterious questions.

In 1996, Mrs. Clinton was ridiculed after Bob Woodward reported, in his book “The Choice,” that as first lady she had held discussions with her deceased role models, Eleanor Roosevelt and Mohandas K. Gandhi. The offbeat tabloid Weekly World News dreamed up sensational headlines about Mrs. Clinton adopting an alien baby and having a “U.F.O. love nest.”

The Clinton presidency also coincided with the hit television series “The X-Files” and movies like “Independence Day,” which gave way to an era of fascination with the existence of aliens and the possibility of a government cover-up.

Mr. Podesta, an X-Files fanatic, ran a fan club for the show in the Clinton White House. “The X-Files fan club would like to invite you and Mulder to lunch at the White House. Don’t let the boss know,” he wrote in a 1998 email, referring to the show’s fictional F.B.I. agents Dana Scully and Fox Mulder, according to White House documents. In 1999, Mr. Podesta had an “X-Files”-themed 50th birthday party that the Clintons attended.

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When Mr. Podesta left the White House last year, before joining Mrs. Clinton’s campaign, he posted on Twitter: “Finally, my biggest failure of 2014: Once again not securing the #disclosure of the U.F.O. files. #thetruthisstilloutthere.”

Mr. Podesta declined to comment for this article.

Mrs. Clinton, who speaks frequently about her childhood aspirations to be a NASA astronaut, has been sympathetic to Mr. Podesta’s efforts.

In 1995, when she was photographed visiting Laurance S. Rockefeller, the billionaire philanthropist, in Jackson Hole, Wyo., she had tucked under her arm a copy of “Are We Alone?: Philosophical Implications of the Discovery of Extraterrestrial Life” by Paul Davies.

Before that meeting, John H. Gibbons, the former director of the White House Office’s of Science and Technology Policy, had warned Mrs. Clinton about Mr. Rockefeller, who had spent years pressuring the government to release files relating to a 1947 crash at a ranch near Roswell, N.M., that had become the source of theories about a cover-up of an alien spaceship. He will “want to talk to you about his interest in extrasensory perception, paranormal phenomena and U.F.O.s,” Mr. Gibbons wrote.

The meeting enthralled conspiracy theorists and, in turn, inspired Hollywood writers.

“If you look at our mythology, there are elements of those kinds of meetings,” said Chris Carter, the creator and executive producer of “The X Files,’’ said in an interview. Mr. Carter, who is supporting Mrs. Clinton, added, “If I have to become a fund-raiser to get an invite to her opening up the files, I’ll do it.”

When Mrs. Clinton started to talk openly about U.F.O.s and government disclosure in her 2016 campaign, some activists traced the remarks back to the 1995 meeting with Mr. Rockefeller.

To this subset of Americans who say the government is covering up what it knows about aliens, and who are incredibly vocal on social media, Mrs. Clinton’s discussion of extraterrestrials signaled an important turn.

Other activists do not care as much about Mrs. Clinton’s vow to “open the files,” but do want prominent politicians to seriously acknowledge that humans may not be the only intelligent life in the universe. A major victory, some say, would be for the candidates to be asked about the topic in a presidential debate.

“It shouldn’t be a source of embarrassment to discuss it,” said Christopher Mellon, a former intelligence official at the Defense Department and the Senate Intelligence Committee.

“We should be humble in terms of recognizing the extreme limits of our own understanding of physics and the universe.”