US President Barack Obama has landed in South Africa, the second stop in his three-country tour of Africa.
Leaving Senegal, he told reporters on board Air Force One: "I don't need a photo op."
Meanwhile, Mr Mandela's ex-wife, Winnie Madikizela-Mandela said the former leader had made "a great improvement" in recent days, but was "still unwell".
But she said she felt it would not be right for the US president to visit Mr Mandela while he was in a critical condition.
Correspondents say security is tight in the streets near Mediclinic Heart Hospital in the capital, Pretoria, where the 94-year old is being treated for a lung infection.
Ministers, politicians, Mr Mandela's physician and family members were among those visiting the ex-leader on Friday, his 21st day in hospital.
Several hundred people earlier took to the streets of Pretoria in protest against Mr Obama's foreign policy.
However, the BBC's James Copnall in the city says Mr Obama does have wide support in South Africa.
'Hero for the world'
Mr Obama's plane landed at a military airbase near Pretoria on Friday evening. During his stay, he is scheduled to meet President Jacob Zuma and hold a major speech, correspondents say. He is also expected to visit Robben Island, where Mr Mandela was imprisoned for 18 years.
However Mr Obama said earlier he did not expect to see the ailing ex-leader during his South African trip.
"I don't need a photo op," he said aboard Air Force One after leaving Senegal. "The last thing I want to do is to be in any way obtrusive at a time when the family is concerned with Nelson Mandela's condition."
He went on to say: "I think the main message we'll want to deliver is not directly to him, but to his family - is simply profound gratitude for his leadership all these years, and that the thoughts and prayers of the American people are with him, his family and his country."
Officials and many South Africans have praised his decision to visit South Africa, our correspondent says.
Mr Obama met Mr Mandela in 2005 when he was still a US senator. Both men became the first black presidents of their nations and have received the Nobel Peace Prize.
The US president has described Mr Mandela as a "hero for the world", whose "legacy will linger on through the ages", and who had inspired his own activism as a student.
'Difficult situation'
Earlier on Friday, Mrs Madikizela-Mandela told reporters outside Mr Mandela's former home in Soweto: "I'm not a doctor but I can say that from what he was a few days ago there is great improvement, but clinically he is still unwell."She also thanked domestic and international media for their coverage, but added that some of the reports had caused the family distress.
"We had no idea of the love for us out there," she said.
"There may be problems here and there when some of you get carried away and talk about our father in the past tense," she said, adding:
"If sometimes we sound bitter, it is because we are dealing with a difficult situation."Earlier this week, Mr Mandela's eldest daughter criticised the international media camped outside the Mediclinic Heart Hospital, calling them "vultures".
There are mixed emotions among the many well-wishers also gathered outside the hospital, the BBC's Karen Allen reports from the scene. People are anxious about his health but also want to express their pride in the man many consider the father of the nation.
"There is no sadness here. There is celebration. He is a giant," 57-year-old Nomhlahla Donry, whose husband served time with Mr Mandela, told the AFP news agency.
"We are saying let him live long because we like him and he has done a lot for us in this world. He's done a lot of amazing things and we really love Tata," local resident Alfred Makhathini told Reuters.
South Africa's ruling African National Congress (ANC) said it would hold vigils each day that the former leader remained in hospital, and the distinctive black, green and gold colours of the party are much in evidence.
But the party denied it was exploiting the occasion to canvas for votes ahead of next year's elections.
"We love our ANC regalia and we have every right to wish Madiba well," party spokesman Jackson Mthembu said.
Meanwhile, a court in the Eastern Cape has granted an application brought by the Mandela family to exhume three of his children and two other relatives and rebury them in the family cemetery in Qunu, which is where the former leader wants to be buried, their lawyers say.
The BBC's Pumza Fihlani in Johannesburg says this is one of the reasons why the family held a meeting earlier this week.
