The reports sparked the #PrayForMenziNgubane hashtag, which quickly rose to the top of Twitter trends on Tuesday.
The family responded to the rumours in an official statement, saying they are aware of the "inaccurate and unverified news" and "unwarranted social media panic."
"We understand their concerns and support for Menzi, but as the family, we appeal to South Africans to refrain from spreading news that has not been verified and confirmed by the family," said the statement issued by Sebabatso Media on behalf of Menzi.
"To this end, we wish to continue as the family to offer him the support he needs following his recent visit to the doctors. It is this visit that has been grossly exaggerated by newspapers and social media users," the statement continued.
"I would like to confirm that my husband is receiving family support and recuperating after doctors were concerned about his health," said the actor's wife, Sikie Ngubane.
The family thanked South Africans for the prayers but asked the Menzi's doctor-patient confidentiality be respected, and said that he would return to work as soon as he has fully recovered.