The lawsuit said: "The companies stole from the Iraqis by engaging in a brazen kickback scheme in which money earmarked for the benefit of Iraqis were instead improperly transferred into the coffers of Saddam Hussein's corrupt Iraqi regime or used to indemnify goods suppliers, including AWB, for the bribes they had paid Iraq."
The lawsuit said the companies disguised and misrepresented the kickbacks to make them appear to be legitimate costs.
It also accused them of knowing or recklessly disregarding that money was being paid illegally to the government of Saddam.
BNP paid as much as $1.5bn in kickbacks to Saddam's government while AWB paid more than $200 million, the lawsuit claims.
Edwina Frawley, a BNP spokeswoman, said she had no immediate comment on Friday.
BNP is headquartered in Paris, France, but has offices in the US, including New York.
"It's the first I have heard of it ... it would be vigorously defended"
A government-commissioned report in Australia was released several weeks ago saying that AWB paid $220m in kickbacks to Saddam's government to secure lucrative wheat contracts under the oil-for-food scheme.
Under the programme, AWB, which had sold wheat to Iraq since 1948, sold 6.8m tonnes of wheat to Iraq, receiving more than $2.3bn in payments, the lawsuit said.
"It's the first I've heard of it," Peter McBride, AWB spokesman said of the lawsuit. "It would be an ill-conceived action, and if it's brought to fruition, it would be vigorously defended."
The lawsuit says the plaintiffs are seeking action in New York in part because the defendants kept offices and did substantial business in the city.
In October 2005, a UN-backed investigation said in a report that about 2,200 companies in the oil-for-food programme, including corporations in the US, France, Germany and Russia, paid a total of $1.8bn in kickbacks and illicit surcharges to Saddam's government.