Airbnb’s situation illustrates a paradox of the start-up dream. Many tech workers join fast-growing privately held companies with the hope of gaining stock in the firms and converting those shares to riches when the start-ups go public. But employees are dependent on the company’s founders and board before that can become a reality.

Mr. Chesky, who co-founded Airbnb in 2008, has been vocal about not rushing to take it public. In January 2018, he published a letter saying the company will have an “infinite time horizon.” He is now exploring a nontraditional initial public offering by potentially listing the shares directly, or on the Long-Term Stock Exchange, which is backed by venture capital but not yet operational, three people with knowledge of the situation said.

Doug Leone, a venture capitalist at Sequoia Capital, one of Airbnb’s backers, said that while start-ups had “an implied social contract” to go public at some point, there was no rush for them to do so. “The I.P.O. is just a moment in time,” he said.

Yet Mr. Chesky’s go-slow stance has become problematic as other high-profile start-ups of the same generation as Airbnb have started listing their shares on the stock market. This year, the ride-hailing companies Uber and Lyft, the online pinboard company Pinterest and the business software maker Slack are among those that have gone public. That has allowed their employees to cash in their shares.

Employee tension is unusual for Airbnb, known for its cheery mission of “belong anywhere” and for fostering a kumbaya culture among its staff. The company has grown rapidly, with more than seven million listings in 100,000 cities. In the second quarter, its revenue exceeded $1 billion. Many employees work out of an airy building in San Francisco, which features rooms that replicate its famous listings. Several former employees said they were grateful for the windfall they would eventually receive from their shares.

Image Dave Stephenson, Airbnb’s chief financial officer, was injured in a ski accident early this year. Credit... via Airbnb

But any reward from owning Airbnb stock has been held back. Starting in 2011, when the young company topped a $1 billion valuation, Airbnb prohibited workers from selling shares, while allowing its three founders — Mr. Chesky, Nathan Blecharczyk and Joe Gebbia — to cash out a total of $21 million.