I don't know that they would be considered rivals any more so than say Bing Crosby and Duke Ellington. I do know that Dylan has said that it was Elvis that made hime want to be a rock and roller, although he'll list Hank Williams, Leadbelly and Woody Guthrie equally high as influences.

There's several obvious differences that set them apart. Elvis was primarily a singer and ultimately a showman, Dylan was primarily a writer who always sought to confound and challenge his concert audience. Elvis embraced celebrity and success, Dylan shunned them. Elvis lived his persoanl life as an open book for fans to share, Dylan got married in secret, twice, never once discussing the state of his personal life in the media. Elvis's legacy is on his recordings, Dylan's is on his song sheets.