Ex-Northern Ireland secretary Lord Hain said that Chancellor Philip Hammond had agreed that the matter would be looked into.
Lord Hain told the Press Association: "I have written to the Chancellor asking for British authorities to investigate British financial institutions that may have been used as conduits for money laundering."
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 Treasury spokesman said: "We take allegations of financial misconduct very seriously, and have passed Lord Hain's letter on to the Financial Conduct Authority and relevant UK law enforcement agencies, including the National Crime Agency and Serious Fraud Office, to agree the right action."
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.