Gray interviewed this week and was a finalist for the position along with LSU running backs coach Larry Porter.
The announcement, assuming there are no unforeseen developments, is not expected to come until after Memphis has played its final game under outgoing coach Tommy West.
Porter played for Memphis and was long considered a favorite for the position.
But Gray was recommended to Memphis by Joe Gibbs, one source said, and had a connection to Fred Smith, the key Memphis booster and founder of FedEx, who is also tied in to the Redskins organization.
Joe Schad is a college football reporter for ESPN.com. Senior NFL analyst Chris Mortensen contributed to this report.