Melnyk, 55, who has been battling health issues for months, was admitted to hospital three weeks ago due to the onset of liver-related complications. His condition is considered critical.
Melnyk released a statement on Monday saying his health became more critical during the past few weeks and that his friends and family convinced him to go public after numerous potential live liver donors were not a match.
More than 500 people have offered to donate part of their liver to Melnyk, and one of his doctors said Saturday that he could undergo surgery by the end of the week.
The Senators announced on Saturday that more than 12 candidates had been identified and selected to go through the donor screening process. The candidates are all at different stages of that process.
A living donor transplant is Melnyk's best option because he has rare AB blood; a live donor doesn't need to have a matching blood type, whereas an organ from a deceased donor must be a perfect match.
"I have been overwhelmed by the outpouring of support from the public and the media. It is truly heartwarming to feel loved by so many people," Melnyk said in the statement, in which he also thanked medical staff at the University Health Network and Toronto General Hospital.
"Words cannot even begin to express how thankful my family and I are for your expressions of support, your prayers, and your unconditional desire to help me in the biggest battle of my life."
Melnyk also stressed the importance of organ donation and the Senators said Saturday that his health was stable and that he is in good spirits.
"You have lifted my spirits, and in the process have given me all the strength I need at this difficult time," Melnyk said. "Thank you from the bottom of my heart."