Loiseau said that in the light of the white paper, France was unclear whether May's government "gets it" on the EU's refusal to haggle over the indivisibility of the four freedoms of the single market - people, capital, goods and services.
The comments come after Mrs May survived a series of knife-edge Commons votes which had the potential to sink her fragile premiership.
"She said she told Mrs May it was"'grossly irresponsible and wreckless" to row back on a commitment to the backstop agreement.
The issue of avoiding a hard border between the Republic of Ireland and Northern Ireland remains the most difficult in the talks.
"It is a matter of urgency to agree on a legally operative backstop for Ireland and northern Ireland".
The EU has proposed a "backstop" solution, to be in place until a satisfactory free-trade deal or technological solution is found, under which Northern Ireland effectively stays in the single market and customs union while the rest of the United Kingdom leaves.
Hugh Savill, director of regulation at the Association of British Insurers, said that even with enhanced equivalence, the EU's "imperialist" approach to rule-making would have a chilling effect on how United Kingdom regulators can supervise markets and consumers.
Parliament voted 307 to 301 against an amendment to trade legislation in a vote that had been closely followed by currency markets, and was seen as a test of May's ability to deliver a smooth British exit from the European Union and keep her job.
The amendment would have required the government to try to negotiate a customs union arrangement with the EU if, by January 21, 2019, it had failed to negotiate a frictionless free trade deal with the bloc.
For example, it says a "no deal" Brexit would "cause significant" delays at borders because of the need to carry out customs checks. He said Ireland will play an active role in supporting the European Union taskforce in its work.
The move raised serious concern in Dublin with the Government stating that a withdrawal agreement can not be agreed by the United Kingdom and European Union unless there is a backstop that has a legal basis.
The EU added, the preparations need to be made for all areas including borders and transport.
Mr Donohoe also said he believes that Mrs May's commitment to an open border is genuine and honest.
Representatives of the EU27 member states will meet in Brussels tomorrow to discuss the state of play in negotiations and the United Kingdom government's White Paper on the future relationship.
Tánaiste and Minister for Foreign Affairs and Trade, Simon Coveney and Minister for European Affairs, Helen McEntee will attend today's gathering to discuss the state of play in the Brexit negotiations.
Britain is set to leave the bloc on March 30, but the two sides want to strike an agreement by late October in order to give parliaments enough time to endorse a deal.
Barnier will spend two hours with ministers from the other 27 member states presenting his views on the white paper, and updating them on his talks on Thursday evening with the Brexit secretary, Dominic Raab.
Nausicaa Delfas, head of global strategy at the Financial Conduct Authority (FCA), told a conference that there were "cliff-edge" risks because of uncertainty over the legality of financial contracts extending beyond the planned Brexit date next March.
Mobile beefs up its unlimited data plans with 70 more countries covered
Well, you've got no choice - they decide FOR you, taking more of your money when you travel internationally. Previously, T-Mobile users could take advantage of free global roaming in 140 territories around the world.
Liverpool Agree £10m Deal To Sell Goalkeeper Danny Ward
Sturridge has inevitably been linked with a move away from Anfield this summer due to his lack of playing time under Jurgen Klopp .
British Open: Tiger Woods shows promising signs in Round 1
McIlroy, who won The Open in 2014, was three shots off the pace after the first round on the Scottish links. The third major championship of the season is underway on a very firm and fast Carnoustie.