The season started a week late in August after the AFE called a stoppage as part of a bid to help players whose wages had not been paid, especially those at clubs suffering from financial difficulties.
A preliminary deal was struck which allowed the season to get under way and the two sides have been negotiating on a variety of issues since then.
However, the LFP said it had now been forced to halt talks in the face of what it described as AFE "intransigence."
"After months of talks, the LFP yesterday felt obliged to break off negotiations over a Collective Accord for professional football given the intransigence of the AFE's position at the table," an LFP statement said.
The AFE published a statement on its website later on Thursday saying the LFP had told the union it wanted a "period of reflexion" and the union could not understand the league's decision to break off talks.
The union added that it was ready to start talks again but suggested an official from the government's sports unit, the football federation (RFEF) or the Olympic Committee (COE) joined them to act as an "independent observer."
"The AFE's proposals have never been set in stone and if on a particular issue there was no agreement they have always waited for an alternative from the LFP, being open to any improvement," the union said.
Among issues on which agreement had not been possible were a new salary structure, players' image rights and disciplinary measures, the LFP said.
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