Mandela's grandchild Zaziwe Manaway reportedly told Eyewitness News that Mandela was "alert".
"He is very alert. My grandfather still wakes up in the morning [and] reads the newspaper so he is also aware of what is being said about him," Manaway told the news station.
"So we'd really like to appeal to everybody out there to stop saying these rumours - they're really aren't true."
Mandela has a history of lung problems, having suffered from tuberculosis towards the end of his 27 years spent in prison.
On Thursday the presidency renewed its plea for the media and public to respect Nelson Mandela's privacy, hours after he was discharged from hospital.
"We call on all South Africans to keep Madiba in their thoughts and prayers and we again appeal for the general public – and especially the media to respect his privacy and allow for an environment conducive to his recovery," Mac Maharaj, presidency spokesperson told the Mail and Guardian on Thursday.
Mandela returned to his Johannesburg home on Wednesday after a 20-day stay in hospital, where he was treated for a recurring lung infection and underwent surgery to remove gallstones.
The 94-year-old continues to receive home based care from a team of specialists monitoring his condition.
Mandela's recent stint in hospital was the longest since being released from prison in 1990.
Maharaj said the media should conduct themselves in a manner that would "assist Madiba's recovery".
"I can't stop the media from doing their job, but what are they hoping to achieve by camping outside his house?"
"Is there a hope they will see him in pain? Do they want him to mistakenly walk out the front gate and have to be led back inside – is that what they want?"
Maharaj said due to Mandela's age, the former president's recovery is "as much physical as it is psychological".
"Anybody coming to the house to visit Madiba should be left alone, so that they should be able to permit as much positive energy to him as possible," he added.
"If someone arrives and they are distressed, what kind of affect can that have on him [Mandela]?"
Maharaj said Mandela is receiving the "best medical attention available", though he would not comment directly on the anti-apartheid campaigner's condition.
Chairperson of the South African National Press Club, Antoinette Slabbert, disagreed with Maharaj though, and said the media had been "restrained up to this point".
"He's a global icon and its naïve for anyone to think the media won't be there," Slabbert told the M&G.
The media were initially misled about the details surrounding Mandela's hospitalisation, leading to suspicions his health was worse than government had reported.
After government inferred Mandela was at One Military hospital in Pretoria, it was later revealed he was receiving treatment at a private medical facility elsewhere in the capital.
While Maharaj or other presidency officials claim to have neither confirmed nor denied Mandela's stay at One Military, defence minister Nosiviwe Mapisa-Nqakula stood outside the facility and claimed to have just visited him.
"While we can't insist on the full details surrounding his care, we would appeal that the communication lines be kept clear so as to inform the South African public about Madiba," Slabbert added..