The 200-pound chimpanzee went berserk in February after its owner asked Nash to help lure it back into her house. The animal ripped off Nash's hands, nose, lips and eyelids.
The hospital, which in 2008 performed the nation's first face transplant but has not done hand transplants, said Monday that Nash has made significant progress in her recovery and more surgeries are planned to further help her regain some independence.
"However, due to the complexity of her injuries, the medical team has concluded she is not a candidate for transplantation at this time," the hospital said in a statement.
The clinic has not ruled out the possibility of some type of collaboration with another hospital, Monaco said.
Nash's family is researching the possibilities of the transplants at a few other hospitals in the United States and one in Canada, Monaco said.
"It will significantly improve her quality of life," Monaco said.
A face transplant would help Nash smell, breath and eat, while a hand transplant would help her be more independent, Monaco said.
Nash has great difficulty eating and mostly uses a straw, he said.
Even if Nash was declared a candidate for the transplants, the surgery would not be done for years, Monaco said.
In April, dozens of doctors working in teams over 30 hours performed the world's first simultaneous partial-face and double-hand transplant at the Henri Mondor hospital in the Paris suburb of Creteil on a 30-year-old burn victim. The man died in June after suffering a heart attack during follow-up surgery.
Dr. Warren Breidenbach, a surgeon at Jewish Hospital in Louisville, Ky., who in 1999 did the nation's first hand transplant, said physical therapy would be extremely difficult for someone who is blind and receives hand transplants.
"When a patient is blind, they cannot feel or see their transplanted hand," Breidenbach said. "The feeling in the transplanted hand takes years to grow back."
The Cleveland Clinic has good hand surgeons, but hand transplant surgery involves a lot of research and a long regulatory process, Breidenbach said.
Prosecutors said in December they would not charge the owner, Sandra Herold, because there was no evidence she knowingly disregarded any risk the animal posed.
Nash, who revealed her heavily disfigured face in November on "The Oprah Winfrey Show," has been at the Cleveland Clinic since soon after the attack. She expects to be discharged soon to an undetermined facility for rehabilitation, Monaco said.
Nash's family is suing Herold for $50 million and wants to sue the state for $150 million. Nash's family has said Herold was negligent and reckless for lacking the ability to control "a wild animal with violent propensities."
Herold's attorney has called the attack work-related - Nash worked for Herold and the animal played a promotional role in Herold's tow-truck business - and said her family's case should be treated like a workers' compensation claim. The strategy, if successful, would limit potential damages and insulate Herold from personal liability.
Test results showed that Travis had the anti-anxiety drug Xanax in his system.
The chimp, which was shot and killed by police, had also escaped in 2003 from his owner's car and led police on a chase for hours in downtown Stamford. No one was injured.