Madoff masterminded a massive and long-running so-called Ponzi investment fraud which came to light in 2008 as the financial crisis gathered speed.
At the time of its collapse in 2008, Madoff Securities claimed it had about USD 65 billion in client assets, whereas in fact it had only about USD 300 million.
The fraud ruined many investors and stoked public anger over the causes and consequences of the crisis.
The Wall Street Journal said: "The bulk of the fines are expected to be routed to victims of Mr. Madoff."
JPMorgan Chase, the biggest US bank by assets, was the main bank used by Madoff for more than 20 years.
The sources quoted by the two papers said the bank had agreed to pay up to close criminal and civil investigations by federal authorities which suspected that the bank had ignored signs that Madoff was operating a fraudulent scheme.
The paper noted that the bank had declined to comment on its report but had insisted that all of its staff had acted in good faith.
The various authorities involved also declined to comment.
The New York Times also reported that in a highly unusual move demonstrating the extreme gravity of the case, the settlement would include a deferred prosecution agreement suspending criminal action provided that the bank acknowledged the facts against it and changed its behaviour.
However, the bank was the target of recent bribery investigations by US authorities over its practise of hiring in China the children of people among the country's ruling elite, the newspaper noted.
In 2009, Madoff was sentenced to 150 years in prison.
His Ponzi or pyramid scheme was based on using money from incoming investors to pay high returns to longer standing investors. When the financial markets collapsed and investors withdrew their funds, the Madoff illusion was exposed.
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