In a farewell television interview, broadcast live on South African television from Qunu, the outgoing President's birthplace in the rural eastern Cape, he said: "Anyone who does not see the white hand [of friendship] is blind or does not want to see it."
President Mandela, who will retire after elections on 2 June, has recently expressed impatience at white inflexibility and a trend among them to emigrate. Some whites believe that his successor, Thabo Mbeki, will pay rather less attention to their concerns.
Last night's interview, which lasted 50 minutes and was jointly anchored by a black and a white journalist, was broadcast live from President Mandela's lounge in his new and apparently rather unlived-in Qunu farmhouse.
The 80-year-old President, wearing a yellow and black shirt, said the first two or three years of his retirement would be dedicated to writing his memoirs.
He said that as a "disciplined and loyal member of the African National Congress" - a phrase that he used several times - he would "listen most sympathetically" to any request for help or advice from future governments of South Africa.
He said he had planted spinach, meelies (corn), onions and carrots on the Qunu farm, as well as acquiring cattle. "I do believe it will be properly organised when I come and live here," said the President.
Even though the interview did not touch on election issues, opposition parties are likely to criticise the national broadcaster, the SABC, for carrying it immediately before the election. The SABC has been accused of pro-government bias recently.
President Mandela said Mr Mbeki would make a very good president. "He has a masters degree in economics. He is a young man with a very favourable political background. During the last two to three years he was the de facto leader and president of this country."
President Mandela said he regarded his party's first five years in power as a time of "unexpected achievement". He said: "We have confounded the prophets of doom and staged a bloodless revolution.
"We have restored the dignity of each South African. In 300 years of occupation no government delivered what we delivered in five years."