The Labour peer, who was a leading anti-apartheid campaigner, raised his concerns in the House of Lords on Thursday.
He will claim that the money-laundering is linked to allegations of corruption involving the South African President, Jacob Zuma, and the Gupta brothers, who set up a series of businesses in South Africa.
Lord Hain said it has been estimated that around £400m may have been laundered through financial institutions in South Africa.
He said the banks may have been used "inadvertently or otherwise" for money-laundering "on an industrial scale".
A spokeswoman for the Financial Conduct Authority said: "The FCA is already in contact with both banks named and will consider carefully further responses received."
In a statement, Standard Chartered said: "We are not able to comment on the details of client transactions, but can confirm that after an internal investigation, accounts were closed by us by early 2014.
"Standard Chartered takes its responsibility to combat financial crime very seriously and is fully committed to doing business in accordance with local and international regulatory and legal requirements."
Mr Zuma has always denied allegations of corruption, as have all members of the Gupta family.
SKY News.