The sale plans were devised by Mr. Wenner’s 27-year-old son, Gus, who has aggressively pared down the assets of Rolling Stone’s parent company, Wenner Media, in response to financial pressures. The Wenners recently sold the company’s other two magazines, Us Weekly and Men’s Journal. And last year, they sold a 49 percent stake in Rolling Stone to BandLab Technologies, a music technology company based in Singapore.

Both Jann and Gus Wenner, the president and chief operating officer of Wenner Media, said they intended to stay on at Rolling Stone. But they said they also recognized that the decision could ultimately be up to the new owner.

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Still, the potential sale of Rolling Stone — on the eve of its 50th anniversary, no less — underscores how inhospitable the media landscape has become as print advertising and circulation have dried up.

“There’s a level of ambition that we can’t achieve alone,” Gus Wenner said last week in an interview at the magazine’s headquarters in Midtown Manhattan. “So we are being proactive and want to get ahead of the curve.”

“Publishing is a completely different industry than what it was,” he added. “The trends go in one direction, and we are very aware of that.”

The Wenners’ decision is also another clear sign that the days of celebrity editors are coming to a close. Earlier this month, Graydon Carter, the editor of Vanity Fair and a socialite and star in his own right, announced he planned to leave the magazine after 25 years. Robbie Myers, the longtime editor of Elle, Nancy Gibbs of Time magazine and Cindi Leive of Glamour also said last week that they were stepping down.

Anthony DeCurtis, a veteran music critic and a longtime Rolling Stone contributing editor, said he never thought Jann Wenner would sell Rolling Stone.

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“That sense of the magazine editor’s hands on the magazine — that’s what’s going to get lost here,” he said. “I don’t know who’s going to be able to step in and do that anymore.”

Wenner Media has hired bankers to explore its sale, but the process is just beginning. BandLab’s stake in the company could also complicate matters. Neither Jann nor Gus Wenner would name any potential buyers, but one possible suitor is American Media Inc., the magazine publisher led by David J. Pecker that has already taken Us Weekly and Men’s Journal off Wenner Media’s hands.

The Wenners said that they expected a range of opportunities, and Jann Wenner said he hoped to find a buyer that understood Rolling Stone’s mission and that had “lots of money.”

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“Rolling Stone has played such a role in the history of our times, socially and politically and culturally,” he said. “We want to retain that position.”

Jann Wenner tried his hand at other magazines over the decades, including the outdoor lifestyle magazine Outside and Family Life. But it was Rolling Stone that helped guide, and define, a generation.

“Who lives through the ’60s, ’70s, ’80s and ’90s and cannot be somehow wistful at this moment?” said Terry McDonell, a former top editor at Rolling Stone who also ran other Wenner magazines.

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Rolling Stone filled its pages with pieces than ran in the thousands of words by standard bearers of the counterculture, including Hunter S. Thompson — whose “Fear and Loathing in Las Vegas” was published in the magazine in two parts — and Tom Wolfe. It started the career of the celebrity photographer Annie Leibovitz, who for many years delivered electrifying cover images, including an iconic photograph in 1981 of a naked John Lennon curled in a fetal position with Yoko Ono.

Music coverage in all of its forms — news, interviews, reviews — was the core of Rolling Stone, but its influence also stretched into pop culture, entertainment and politics. A bastion of liberal ideology, the magazine became a required stop for Democratic presidential candidates — Mr. Wenner has personally interviewed several, including Bill Clinton and Barack Obama — and it has pulled no punches in its appraisal of Republicans. In 2006, Rolling Stone suggested George W. Bush was the “worst president in history.” More recently, the magazine featured Justin Trudeau, the prime minister of Canada, on its cover with the headline, “Why Can’t He Be Our President?”

The magazine also published widely acclaimed political stories, including one in 2009 on Goldman Sachs by the writer Matt Taibbi, who famously described the company as “a great vampire squid wrapped around the face of humanity.” The next year, the magazine ran a piece with the headline, “The Runaway General,” that ended the career of Gen. Stanley A. McChrystal.

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But that was perhaps the last Rolling Stone cover piece that gained significant journalistic acclaim. And the magazine’s reputation as a tastemaker for the music world had long since eroded, as Mr. Wenner clung to the past with covers that featured artists from his generation, even as younger artists emerged. Artists like Paul McCartney, Bruce Springsteen and Bob Dylan have continued to secure cover spots in recent years.

Rolling Stone suffered a devastating blow to its reputation when it retracted a debunked 2014 article about a gang rape at the University of Virginia. A damning report on the story by the Columbia Graduate School of Journalism cited fundamental journalistic failures. The article prompted three libel lawsuits against Rolling Stone, one of which led to a highly publicized trial last year that culminated with a federal jury awarding the plaintiff $3 million in damages.

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The financial picture had also been bleak. In 2001, Jann Wenner sold a 50 percent stake in Us Weekly to the Walt Disney Company for $40 million, then borrowed $300 million five years later to buy back the stake. The deal saddled the company with debt for more than a decade, preventing it from investing as much as it might have in its magazines.

At the same time, Rolling Stone’s print advertising revenue and newsstand sales fell. And as readers increasingly embraced the web for their news and entertainment, Mr. Wenner remained skeptical, with a stubbornness that hamstrung his company.

Wenner Media was already a small magazine publisher. But the sale of Us Weekly and Men’s Journal, which together brought in roughly three-quarters of Wenner Media’s revenue, has left it further diminished.

Regardless, the sale of Rolling Stone would be Jann Wenner’s denouement, capping his unlikely rise from dope-smoking Berkeley dropout to silver-haired media mogul. An admirer of John Lennon and publishing mavens like William Randolph Hearst, Mr. Wenner — who invested $7,500 of borrowed money to start Rolling Stone along with his mentor, Ralph J. Gleason — was at turns idealist and desperado, crafting his magazine into a guide for the counterculture epoch while also gallivanting with superstars. He once boasted that he had turned down a $500 million offer for Rolling Stone, more than he could ever dream of getting for the magazine today. (BandLab invested $40 million to acquire its 49-percent stake in the magazine last year.)

Though he said he still cared deeply about Rolling Stone, Mr. Wenner has placed the magazine’s fate firmly in Gus’s hands, and he appears content to let someone else determine its path forward.

“I think it’s time for young people to run it,” he said.

Sitting in his second floor office surrounded by a collection of rock ’n’ roll artifacts, Gus Wenner expressed hope that a new owner would provide the resources Rolling Stone needed to evolve and survive.

“It’s what we need to do as a business,” he said. “It’s what we need to do to grow the brand.”

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Then, as only someone who had spent his life around rock ’n’ roll could, he gestured confidently to a tome of Bob Dylan lyrics on his desk. “If you’re not busy being born,” Mr. Wenner said, “then you’re busy dying.”