John Codey, a friend of the late tycoon who runs the fund charged with looking after the white Maltese named Trouble, said between 20 and 30 threats had come in.
People reportedly told him: "I'm gonna kill the dog," and "I'm gonna kidnap the dog. I need the $12 million."
Leona Helmsley, who owned several landmark luxury hotels in New York along with numerous properties throughout the United States, died on August 20 at the age of 87.
Although dubbed the "Queen of Mean" and jailed for tax evasion, she reportedly made substantial contributions to charitable causes, including multi-million dollar donations to hospitals and disaster relief work.
During her tax evasion trial, a former housekeeper famously quoted her as saying: "only the little people pay taxes."
Her will, which was made public in court soon after her death, grabbed headlines in the U.S. due to its highly unusual provisions. While leaving millions to a fund for the upkeep of her dog and $3 million for maintenance of her $1.4 million mausoleum, she disinherited two of her four grandchildren, citing "reasons which are known to them."
Leona Helmsley left $10 million to her 80 year-old brother, Alvin Rosenthal, and charged him with managing Trouble's fund, but he refused.
The other two grandchildren by her late son Jay Panzirer received $5 million each on the condition that they visit their father's grave at least once a year.
In her will, the billionaire wrote: "I direct that when my dog, Trouble, dies, her remains shall be buried next to my remains in the Helmsley mausoleum." The lavish mausoleum also contains Helmsley's husband Harry, who died 10 years ago.
However, the administration of the cemetery in Westchester County has said burying the dog with the couple will be impossible.