Suspicious activity reports (SARS) leaked from US Financial Crimes Investigation Network (FinCEN), a US Treasury agency focused on money laundering, indicated US$2trn of transactions through the global banks between 2011 and 2017.
Buzzfeed, which received the leaked documents along with the International Consortium of Investigative Journalists (ICIJ), said the documents showed how groups such as the Taliban, Russian oligarchs and gangsters moved money through the western bank system.
SARS are documents written by internal officers at the banks and sent to FinCEN and do not, in themselves, indicate wrongdoing.
FinCEN had warned two weeks ago that there had been an illegal leak but subsequently has asked for comments on how the US can improve its anti-money laundering procedures.
HSBC said the leaked documents related to historical events. They also pre-date an agreement with the US authorities following a failure to halt Mexican drug cartels laundering money through the bank.
HSBC told the BBC it has always met its legal duties on reporting suspicious activity.
Standard Chartered added: "The reality is that there will always be attempts to launder money and evade sanctions; the responsibility of banks is to build effective screening and monitoring programmes to protect the global financial system."
"We take our responsibility to fight financial crime extremely seriously and have invested substantially in our compliance programmes," said the emerging markets-focused bank.
Shares in HSBC fell 4% to 291.9p and Standard Chartered dropped 3.9% to 345.6p.