“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 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 rooted more in science fiction than in science, 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 also a crusader for the 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.