The modest, four-bedroom house at 1034 Audubon Drive was posted on eBay on Friday.
As of last night, the lone bid was for $100,000 (€81,000) by Elvis Unlimited, an organisation that operates an Elvis museum and shop in Randers, Denmark.
Presley, then 21 years old, bought the home on March 8, 1956, with his royalty earnings.
The singer, his parents and grandmother lived in the house for only a year.
A month after they moved in, Heartbreak Hotel hit No 1 on the Billboard charts, ultimately ending any privacy he had in the neighbourhood.
Crowds of fans lined the suburban street and police frequently had to be called in to handle the problem.
A Life magazine article from August 1956 had pictures of teenage girls sitting with their ears pressed to his bedroom wall and picking through the grass in his yard for souvenirs. The commotion became so intense that Elvis moved his bedroom to the back of the house.
In the short time the family was there, Elvis had a 50-foot granite pool installed in the backyard and a den added to the house, according to records. The pool was the largest residential pool in Memphis when he built it.
His motorcycles were housed in a separate building, which eventually became a pool house complete with two dressing rooms. Both the pool house and the pool remain on the property.
Many of Alfred Wertheimer's photographs of the emerging musician were shot at the house, chronicling Elvis' growing fame and fortune.
In March 1957, Presley gave up the house on Audubon Drive for a 14-acre estate with a two-story colonial house already known as Graceland.