LOS ANGELES — God moves in mysterious ways — particularly when it comes to Los Angeles real estate.

And few transactions have proved more mystifying of late than conflicting attempts by the Roman Catholic Archdiocese of Los Angeles and a group of nuns under its supervision to sell an eight-acre hillside estate here.

The property might go to the pop singer Katy Perry. Or it could fall to the restaurateur and developer Dana Hollister, depending on whether the sacred matchup or a more profane alignment of lawyers and would-be buyers prevails.

Ms. Perry — a gospel singer turned pop star whose early secular hits included “Ur So Gay” and “I Kissed a Girl” — is offering an all-cash deal favored by the archdiocese, which says it controls the property. Ms. Hollister, who helped open trendy Los Angeles restaurants like Cliff’s Edge and Villains Tavern, is offering a potentially richer deal favored by the nuns but complicated by its reliance on a large promissory note.

Some on both sides have been shocked by the highly public nature of the fight, which has spilled into court and onto television screens. It pits the local archbishop, José Gomez, against a pair of nuns, Sister Rita Callanan, 77, and Sister Catherine Rose Holzman, 86 — both appeared on NBC’s “Today” show this week — who are among five remaining members of a Los Angeles order, the Sisters of the Most Holy and Immaculate Heart of the Blessed Virgin Mary.