Last month, Stars Earn Stripes producer David Hurwitz claimed Jack was never hired for the tough reality contest.
He said: "He was somebody we were in talks with."
Both Sharon and Jack - who was first contacted about the show in April - admit a contract was never signed but insist the deal was moving forward on a verbal commitment.
After he agreed to take part, the Adrenalin Junkie host - who went public with his debilitating medical condition in June - received an email on May 18 from Chuck LaBella, the network's vice president of talent relations, confirming his participation.
However, soon after disclosing his condition, Sharon says she received an email from NBC, which read: "We cannot move forward with Jack's participation".
She also claims NBC offered to pay Jack - who said he would fund his own insurance policy - his full appearance fee but they declined.
A representative for NBC said they regretted any "misunderstanding" which has led to Sharon leaving America's Got Talent.
The representative said: "Sharon has been a valuable part of the NBC family. We regret any misunderstanding and wish Jack well."
- Bang! Showbiz