Serge Blanco, president of the Ligue Nationale de Rugby, has blamed the RFU for failing to hand over half of its shareholding and voting rights to English clubs.
The French also argue there is no room to squeeze in European Cup fixtures because their own domestic league will not start until after the World Cup final on October 20.
The current Paris Accord, the six-nation agreement on how the tournament is run, expires this summer.
A new agreement is "effectively ready" but this row has delayed its signing.
There is a meeting of the ERC shareholders on February 7, which will be crucial to the outcome of this latest club-versus-country rumpus.
But Lux called on the RFU and Premier Rugby to make the first breakthrough.
He said: "For the moment, we are waiting. Serge Blanco cannot change his position. But what can change is the situation in England between the clubs and the union.
"The English and French clubs want to have power in the tournament. When Bernard Lapasset (president of the French union) agreed with Serge Blanco to give him 48% of the shareholding, Serge was happy with that.
"But he was waiting for the same in England. With English teams and French teams having those shares, he thinks it will be better for control of the ERC in the future."
If the French clubs pull out of the tournament, it is possible the English clubs could follow suit.
Asked if he is confident a full-strength tournament will take place next year, Lux indicated it was far from certain.
He said: "It is a board decision. We have a meeting next Wednesday and, of course, we will talk about that.
"For the moment, I cannot tell you if everybody will agree."
ERC chief executive Derek McGrath added everything would be done to try and ensure a full-strength tournament will be held - but he urged the French clubs not to become too entrenched in their position.