Lance Armstrong's broken collarbone may not improve his chances of winning the Tour de France this year. But his aspirations for the race have gone farther than a record eighth victory.

In 2006, Mr. Armstrong embarked on an effort to overhaul professional cycling that, in some scenarios, would involve him owning a stake in cycling's most prestigious race. The race is owned by a family-run French company that says it has been approached by numerous entities looking to buy the race over the years. It says the Tour de France...