The New York Times reports that the body called Parquet National Financier is concerned the Paris-based federation approved the sale of the sport to Liberty despite holding a stake.
The FIA denies it was a conflict of interest.
“We are entirely confident that any investigation would find that the FIA has acted appropriately at all times, and we stand ready and willing to cooperate with any enquiries should any investigation be commenced or clarification sought by the appropriate authorities,” the governing body said.
Britain’s Serious Fraud Office is also investigating the sale.
The FIA’s 1 per cent stake in the sport was agreed in 2013 between president Jean Todt and former F1 chief executive Bernie Ecclestone.
“This is what they wanted in order for them to go along with it,” Ecclestone said.
“A lot of things happen in business that if you look at it you can’t understand why it is allowed to happen. Somebody said to me there’s a problem.
“I think in the end people were part of it in France and they’ll have a look at these things if they think it’s wrong,” he added.
A spokesperson for Liberty did not comment.