Presley was not Jewish. Yes, some sources repeat a tall tale that Presley's third cousin told 20 years ago to a Jewish biographer of Presley. This cousin said that he and Presley shared a very remote Jewish maternal ancestor (a woman who lived in the early to mid 1800s).
This biographer did no further checking on this cousin's story. She just reported it as "fact."
A detailed check of available records, including the census, shows that this maternal ancestor was not Jewish. Presley and his parents did share a two-family house in Memphis, Tennessee with a poor Orthodox rabbi and his family in the early '50s.
I spoke to the adult daughter of this rabbi. Sadly, her father, the rabbi, died young. She was a child at the time of his death and she never talked to him about Elvis.
She told me that her mother, the rabbi's wife, and Presley's mother, Gladys, were close friends. Gladys, her mother told her, never said a word about any Jewish ancestor. Her mother said that if Gladys had any knowledge of any Jewish ancestor, the mother was sure Gladys wouldn't have hesitated to mention it.
Presley worked for the rabbi's family, doing tasks Jews were not permitted to do on the Jewish Sabbath. He did this for free. The rabbi, in turn, did things like lend Presley his record player and arrange for a summer camp trip for Presley. When Presley hit it big, he made a major donation to the rabbi's religious school.