His mother, on her deathbed in 2002, told Tim again that Elvis was his father. From that point on, Tim began seeking DNA evidence to determine his parentage. At the time of this writing, he has been unable to obtain usable DNA to prove or disprove his mother's claims. In any case, Tim does bear some resemblance to Vernon Presley, Elvis' father, and some say Tim's singing voice is reminiscent of Elvis'.
She grew up hearing stories of how her mother had met Elvis in North Carolina during the 1950's but she never considered that the famous singer might actually be her father until someone mentioned that she greatly resembled him. She soon began to wonder and eventually asked her mother if Elvis Presley was her father. Deborah's mother said that he was.
In the 1980's Deborah went to court to have herself named a legal heir of Elvis Presley. The court found that she was not Elvis' daughter and could not, therefore, claim any part of his estate. An appeals court, however, ruled that the prior ruling regarding paternity was invalid. The appeals court did not say that Deborah was definitively Elvis' daughter--only that there had not been enough evidence to rule that she couldn't be. The appeals court did agree that in either case, Deborah had no right to any part of Elvis' estate.
The factual veracity of the book (aptly named, Are You Lonesome Tonight?), has been called into question by both friends and fans of Elvis.
It is interesting to note that Desiree does resemble Elvis quite a bit. While that is certainly not proof of kinship, it is another factor to consider.