Mandela was released from the hospital 25 days ago after spending nearly three months in treatment for a lung condition.
Back at home in Houghton, his family said the 95-year-old former leader appears "vibrant" and was acknowledging their presence.
"I just saw him again yesterday at lunch. He is doing very well," grandson Mbuso Mandela said Wednesday.
He said Mandela was not completely bed ridden, The (Johannesburg) New Age reported.
South African President Jacob Zuma told the U.N. General Assembly in New York that Mandela was responding well to treatment at his home.
"His family and our people have warmly welcomed the support he continues to receive from the global community," Zuma said.