Fifa said that a former finance committee chief who died last year, Argentine Julio Grondona, authorized the payment which went to Jack Warner.
Warner, one of 14 people facing corruption charges in the United States over $150 million in bribes, was at the time Grondona's deputy as head of Fifa's finance committee.
The South African government asked FIFA to “withhold” money intended for the organizers of the 2010 World Cup and send it to a development project in the Caribbean run by Warner, a Fifa statement said.
According to US investigators, the $10 million was a bribe promised to Warner and his deputy Chuck Blazer to secure the 2010 World Cup.
South Africa has strongly denied any wrongdoing.
Corruption battle grows
The Fifa statement came in response to a New York Times report that Valcke, right hand man to Fifa president Sepp Blatter had approved the payment to an account controlled by Warner, then head of the North and Central American and
Caribbean confederation (Concacaf).
The New York Times, citing unnamed law enforcement officials, said the payments are a crucial part of the indictment against the 14 football officials and marketing executives.
Fifa said it had acted as an intermediary between South Africa and a World Cup legacy project to “support the African diaspora in Caribbean countries”.
A Fifa statement said neither Valcke “nor any other member of FIFA's senior management were involved in the initiation, approval and implementation” of the transfer.
Swiss authorities said that they have received no US request for assistance regarding Valcke, a Frenchman.
Valcke on Monday announced that he had called off a planned visit to Canada next week for the start of the Women's World Cup.
Blatter still plans to attend the final.
“The payments totalling $10 million were authorized by the then chairman of the finance committee and executed in accordance with the organization regulations of FIFA,” added the statement.
Grondona, a controversial figure in world football over many decades, was finance committee chief at the time.
He was a senior FIFA vice president when he died last year at the age of 82.
Swiss police, following a US request, arrested seven Fifa officials in Zurich two days before Blatter was reelected as president on Friday.
Blatter, who has faced calls, even since his reelection, to stand down, has expressed doubts about the raid.
But the corruption scandal has since spread with authorities in several countries launching investigations.
Brazilian prosecutors have launched an investigation into Ricardo Teixeira, former Brazilian Football Confederation (CBF) president, for money-laundering and fraud, officials said Monday.
Jose Maria Marin, who was CBF president until April, was one of the seven people detained in Zurich.
Texeira led the CBF for 23 years until 2012. Federal government spokesman Marcelo del Negri, confirmed to AFP that Texeira was under investigation.
A police probe found that between 2009 and 2012, Texeira moved 464 million reais (about $150 million) through his accounts — an amount that drew attention as “unusual,” the newsmagazine Epoca reported after gaining access
to the police file.
Teixeira, now 67, stepped down in 2012 amid corruption allegations.
He is the son-in-law of former Fifa president Joao Havelange, who was also involved in multiple controversies while head of the world body.