Back when America Online was a dial-up internet giant, the conference now known as the Pac-12 was a men’s basketball power.

Arizona and UCLA each won a national championship in the 1990s, and conference teams made eight Final Four appearances in the 1990s and 2000s. But two years after Oregon’s appearance in the 2017 Final Four—the conference’s only one since expanding from the Pac-10 in 2011—the sunny Pac-12 is having its darkest and stormiest season.

As the Pac-12 tournament tips off Wednesday, the conference of UCLA coach John Wooden’s 10 titles in the 1960s and ’70s is at risk of qualifying just a single team for the NCAA tournament. (Washington won the regular-season title, but only the winner of the conference tournament gets an automatic bid.) It would be the first time that’s happened to one of the nation’s most powerful conferences since the tournament expanded to 64 teams in 1985.

“I suspect they’ll end up getting two teams,” said Ken Pomeroy, founder of the basketball analytics site kenpom.com, “but obviously it’s a dire case when you’re even questioning that.”

Not only are conference kingpins struggling, they’re scandalized. Conference titan UCLA is looking for a permanent coach after firing Steve Alford in midseason following a 15-point loss at home to Liberty. Last season, three Bruins players were arrested for shoplifting in China—while on a trip to raise the Pac-12’s profile abroad, no less. The incident spurred U.S. government intervention and tweets from President Trump.

Former Arizona assistant coach Emanuel “Book” Richardson pleaded guilty in January to accepting illicit payments for steering athletes to certain business managers. The plea was part of a continuing federal case involving multiple schools that’s still hanging over the Wildcats and coach Sean Miller.

A continuing federal case is still hanging over the Arizona Wildcats and coach Sean Miller. Photo: Christian Petersen/Getty Images

The Pac-12’s problems extend beyond its former Big Two. A tempest of coaching turnover, the loss of 21 players to the NBA Draft after the past two seasons and key injuries all have hurt the conference.

The Pac-12 is on track to finish the season having spent 11 weeks with zero teams in the AP top 25 poll. The only worse showing since the poll’s expansion in 1989-90 was the 15 weeks of exile in 2011-12—also by the Pac-12. In this week’s poll, the conference didn’t receive a single vote.

“Basketball is cyclical,” said Phil DiStefano, University of Colorado chancellor and chairman of the Pac-12 board of directors. “We’ve seen success just a couple of years ago,” including having three teams in the Sweet 16 in 2017, he said.

The Pac-12’s celebrated moniker as the conference of champions recently has rested on winning NCAA titles in Olympic sports such as water polo, rowing and volleyball.

The four-team College Football Playoff has selected a Pac-12 team twice in five years of existence. The conference hasn’t won a football national championship since the University of Southern California won back-to-back titles in 2003-04 (though the latter was vacated by the Bowl Championship Series for NCAA violations).

Some leaders at Pac-12 schools worry that the struggles in football and men’s basketball could worsen with schools’ growing financial challenges, which in turn could make it more difficult to sign or re-sign top coaches and assistants.

Lagging revenues from the seven-year-old Pac-12 Network have contributed to the conference falling behind. In 2016-17, each Pac-12 school received $30.9 million in conference revenue, while schools in the top-dog Southeastern Conference each got $40.9 million.

The Pac-12 always has suffered from time zones unfriendly to East Coast audiences and local fan bases with palm trees and pro teams to distract them. But in recent years, the conference also has been hurt by its own far-reaching vision.

The decision to launch a network without an established media partner, made by commissioner Larry Scott and Pac-12 school presidents and chancellors, gave the conference more control. But the network has proved costly to operate and difficult to sell to carriers.

The revenue gap between the Pac-12 and other power conferences is widening and could continue to grow until 2024, when its $3 billion primary-rights deals with Fox and ESPN expire and new ones are expected to fetch another big payday.

Arizona State and coach Bobby Hurley are on the tournament bubble. Photo: Rick Scuteri/Associated Press

“We’re concerned,” Stanford athletic director Bernard Muir said of his conference peers. “We want to stay in the game. We want to be competitive.”

The Pac-12 is exploring, with the help of investment bank the Raine Group, an unusual move to improve its financial position. The plan would bundle all the conference’s media and sponsorship rights and sell a stake in them, perhaps 10% to 15%, to an investor. The resulting up-front windfall—$500 million to $750 million, by the Pac-12’s reckoning—would help schools bridge the gap until 2024.

The conference’s hope also is that the right partner could bring expertise to better manage Pac-12 rights and increase their overall value. But conference leaders also could opt for no deal, and wait out the next five years.

The Raine Group is scheduled to update the Pac-12 board and athletic directors on Saturday.

“At one point we were ahead and then all these deals got re-upped by other conferences and then we slipped behind,” Muir said. “Hopefully we’ll have an opportunity somewhere down the road to close the gap.”

Write to Rachel Bachman at rachel.bachman@wsj.com