Lord Peter Hain has raised concerns the banks may "inadvertently have been conduits" for laundered money.
The Labour peer is due to raise the issue in the House of Lords later.
The Treasury said it had sent Lord Hain's comments to financial regulator the Financial Conduct Authority (FCA).
Lord Hain told the FOW24 he had written to Chancellor Philip Hammond saying a whistle-blower had indicated the banks "maybe inadvertently have been conduits for the corrupt proceeds of money". The peer is due to raise the issue in the House of Lords later.
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 spokesperson said.
Lord Hain's letter was "a new twist in a giant corruption scandal that is shaking the South Africa state, and damaging the reputations of a number of global companies".
At the centre of the allegations are South Africa's President Jacob Zuma and a wealthy business family, the Guptas.
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Zuma and the Guptas - the Zuptas
Mr Zuma and the Guptas strongly deny wrongdoing, and say they are victims of a "politically motivated witch-hunt".
But leaked emails and official investigations have fuelled claims that the Guptas have bought influence in government in order to loot state enterprises.
In South Africa, the scandal has already ruined British public relations company Bell Pottinger and damaged auditors KPMG, which removed its top executive team in the country.
Bell Pottinger goes into administration
KPMG clears out South African management
A spokeswoman for the FCA said it was already in contact with the banks named by Lord Hain and would "consider carefully further responses received".
Standard Chartered said: "We are not able to comment on the details of client transactions, but can confirm that following an internal investigation accounts were closed by us in 2014." HSBC declined to comment.
Lord Hain, a leading anti-apartheid campaigner who grew up in South Africa, urged UK authorities "to track that stolen money down and make sure that British financial institutions help return it to South African taxpayers".
It is claimed that money was taken out of South Africa via Hong Kong and Dubai.
Lord Hain, a former Northern Ireland secretary, alleged in his letter to the chancellor that the issue was "a result of the corruption and cronyism resided over by President Zuma and close allies the Guptas".
Brothers Ajay, Atul and Rajesh Gupta have interests in computer, mining, media, travel, energy and technology and employ about 10,000 people through their company Sahara Group.
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