Mandela, who is on life support, is not in a vegetative state, the source said.
The anti-apartheid icon has been hospitalized for nearly a month for a recurring lung infection.
Considered the founding father of South Africa's democracy, Mandela became an international figure while enduring 27 years in prison for fighting against apartheid, the country's system of racial segregation. He emerged from prison in 1990 and became the nation's first black president four years later.
His lung problems started during his years in prison under South Africa's now-defunct apartheid regime.
He was admitted to a Pretoria hospital on June 8, and since then, varying accounts of his medical condition have emerged.
His health had declined so sharply last week that his family was considering whether to take him off life support before his condition improved, a court document released Thursday revealed.
The document, known as a certificate of urgency, was filed by his family on June 26 in a burial dispute.
It stated that Mandela, 94, had "taken a turn for the worst and that the Mandela family have been advised by the medical practitioners that his life support machine should be switched off."
"Rather than prolonging his suffering, the Mandela family is exploring this option as a very real probability," it added.
The following day, however, South African President Jacob Zuma announced that Mandela's condition had improved from critical to critical but stable.
Another document filed by Mandela's family in the burial dispute described his health as "perilous" and stated that fears that his death is drawing near were justified.
Mandela remained in critical but stable condition Thursday, Zuma's office reported after he visited Mandela. It denied reports that Mandela was in a "vegetative state."
The court battle pitted 16 members of Mandela's family against his grandson, Nkosi Zwelivelile Mandela, also known as Mandla, over where three of the anti-apartheid icon's deceased children should be buried.
Mandla Mandela lost his case Wednesday, which meant the return of the remains of the three relatives to the family graveyard in Qunu, Nelson Mandela's boyhood home, could go ahead.
They were reburied Thursday in the family compound.
But the dispute brought a public chiding from another hero of the anti-apartheid cause, retired Archbishop Desmond Tutu.
In a statement carried by the South African Press Association late Thursday, Tutu urged the family not to "besmirch" Mandela's name in his last days.
Investigators have been gathering evidence against Mandla Mandela in a "grave tampering" case and expect to hand it to a senior prosecutor next week, Mthatha police said Friday, according to the South African Press Association.
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