The proposal has been backed by the European Parliament and Commission, as well as some member states, including the UK.
They want to see a change to the current system, in which hedge funds from outside the EU have to seek the backing of individual member states if they want to sell in their markets, reports the Financial Times.
But the French concession has come at a price - the country wants strict conditions to be introduced, with the European Securities and Market Authority put in charge of issuing passports and regulating hedge funds.
"Our position is not protectionism but has always been about ensuring maximum protection for investors," a French official said.
Earlier this year, US Treasury secretary Tim Geithner said he was concerned about some aspects of the proposals and wanted to see hedge funds based outside the EU given the same market access as their European counterparts.
By Claire Archer