Gregory, the bishop of the diocese of Bellville, Ill., became the first African-American elected to lead the U.S. bishops' conference in 2002 and guided the church through reforms after years of scandals over sexual abuse by priests.
George is opposed by SNAP, the Survivors Network of those Abused by Priests, a support group for clergy sex abuse victims, which backs Bishop Blasé J. Cupich of Rapid City, S.D., the Chicago Sun-Times said.
SNAP also opposes conference vice chairman Bishop William Skylstad of Spokane, Wash., considered the frontrunner for the post.
The bishops' conference meets in Washington Monday.
Celebrity Families of 2014 [PHOTOS]