For weeks, opinion polls have indicated the majority of French voters intend to say "no" to the constitution. The latest poll indicates the size of the negative vote is declining, but it's still ahead.
Another EU stalwart, the Netherlands, appears also to be wavering, with the rejectionists forging ahead in one of the latest opinion polls. So what is happening to the European project?
Senior analyst Philippe Moreau de Farge, of the Paris-based French Institute for International Relations, told RFE/RL that many French see the liberal economic philosophy written into the constitution as promoting globalization. That's a free-trade system that is seen as undercutting wages and benefits in the developed world.
"There is a confusion, a comparison between the EU and [economic] globalization, given that the economic and social situation in France at present is bad. And in the end result, Europe is seen as [the vehicle for] this globalization," Moreau said.
Moreau de Farge said there are many other reasons why the French public is concerned about the constitution, not least of them being Turkey's bid to join the EU. That prospect is not popular in France, with many believing that Turkey is too big and too culturally different to join Europe.
"The 'no' votes can be explained by many, many different motives, and certainly Turkey is one of them. Some French seem to be convinced that with the constitution, Turkey could join the European Union without any kind of [proper membership preparation] proceedings. Of course, that idea is wrong," Moreau said.
The French are also said to be trying to punish the government of Prime Minister Jean-Pierre Raffarin for the country's stagnant economy, for moving to dismantle the 35-hour working week, and also for the continued immigration problem.
Politicians from around Europe are urging the French people not to reject the constitution, which is meant to streamline EU procedures so as to avoid paralysis in the enlarged 25-nation bloc. One of the document's supporters is current EU External Trade Commissioner Peter Mandelson, who said rejection would not necessarily mean the EU would fall apart, but that it could "face some sort of chaos."
French Foreign Minister Michel Barnier also had a dire warning. "Every country that would refuse this [EU Constitution] text is weakening itself, is isolating itself from the rest," he said. "This would be even more so for a founding country such as France. Their influence would be crippled. You have to know that a lot of our proposals and ideas have been included in the constitutional project. Moreover, not only the country in question would be weakened, but the European Union as a whole."
Analyst Moreau de Farge is nevertheless guardedly optimistic. "The French can be crazy, but they can also be reasonable, and nothing is yet settled [on the question of the constitution]," he said.
The Netherlands looks like a similar case to France. A founding member and enthusiastic supporter of the EU, the Netherlands seems to be going through a period of uncertainty. Some of the latest opinion polls show the prospective "no" voters in the lead, others the reverse.
Some analysts link the change in the country's previously ultraliberal mood to the murder several years ago of a rightist politician who opposed immigration, and also the killing last year of a filmmaker who was critical of Islam.
So far, six countries, including Italy and Spain, have ratified the constitution. In most cases, there is little prospect of failure as parliaments will ratify the treaty. But in the half-dozen countries where referendums are still to be held, the road ahead could be bumpy.