Fox and Apollo would have a 50-50 stake in the venture and jointly manage it.
The aim is to broaden distribution of hit shows around the world while funding the creation of new ones. There are already versions of "Idol" in more than 100 countries.
Other shows made by the companies include "So You Think You Can Dance," ''MasterChef" and "Deal or No Deal."
It's not immediately clear what will happen with Elisabeth Murdoch, who is Shine's chairman. She is also the daughter of Rupert Murdoch, who controls Twenty-First Century Fox Inc. through a family trust that owns shares.
News Corp., which once housed Fox before splitting into two companies in June last year, bought Shine for about $480 million in April 2011.
The agreement comes after the Fox broadcast network cut back hours of "American Idol" in the U.S. after the current season was weak with audiences. However, Fox entertainment chief Kevin Reilly told advertisers this week that he believed "Idol" could be a solid performer in the U.S. "for many years to come."
Shares of 21st Century Fox slipped 26 cents to close at $34.02 Thursday amid a broad stock market decline.