In November 2008, two officials from the International Golf Federation traveled to Switzerland to make a case for returning their sport’s athletes to the Olympics after a century-long hiatus. But as the men made their presentation to members of the International Olympic Committee, they quickly realized they were pitching to an audience fixated on a single player.

Never mind the sport’s global reach or its charitable impact and cornerstone values of honesty and fair play. To Ty Votaw, who was helping make golf’s case for inclusion, it was clear from the start what the I.O.C. considered golf’s greatest asset.

“The very first question asked was, ‘Will Tiger Woods play?’” Votaw said.

Twelve years later, the Olympics still covet Woods — now, perhaps, more than ever. The problem is that there may not be a spot for him.

In 2008, Woods’s participation seemed foreordained. He stood alongside Usain Bolt and Michael Phelps as one of the world’s most dominant athletes, and was coming off a season that had included his 14th major title. For the 10th time in 11 years, he ended the year ranked No. 1. Woods, with his star power, and the Olympics, with their global stage, looked like a marriage brokered in a corporate boardroom.