Scheduled for republication in the US by the Thomas Nelson group on 3 June, is the biography: "Anchored In Love: An Intimate Portrait of June Carter Cash". Written by John Carter Cash, the son of Johnny and June Carter Cash, the biography includes a very interesting story about the relationship between Elvis and June:
Throughout my life, I would see Mom get a mischievous twinkle in her eye whenever she mentioned Elvis Presley. Her eyes would flash merrily, and she would say, "you know, son, your father was always jealous of Elvis." She even told me once that she sometimes wondered what would havew happened if she had fallen in love with Elvis.
Mom and Elvis occasionally toured together, along with other performers, sometimes including Mother Maybelline and one or more of June's sisters. The Carter's were friends of Elvis, and there are stories about Mother Maybelline sewing buttons on Elvis' shirts when they popped off during his wild onstage gyrations.
Though Mom always maintained that she never had an affair with Elvis, Carl [her first husband] believed differently and perhaps for good reason. After Carl moved out of their Madison home, Mom would sometimes let Elvis stay at the house to "rest" after a tour.
Source: elvisinfonet.com