Strike postpones start of Spanish season
"Both parties have explained their positions and have agreed to continue with the conversations over the meetings already agreed, on Saturday at and on Monday at the same time," the LFP said in a statement.
The AFE are demanding greater protection for players' wages at clubs who have gone into financial administration and last week called a strike for the first two rounds of matches in the top two leagues.
The AFE say around 50 million euros is owed by clubs to more than 200 players, and that a proposed emergency fund of 40 million euros put forward by the LFP was not enough.
The first meeting between the sides on Wednesday ended quickly with no resolution.
"I prefer not to speak about what may happen," said Jose Luis Astiazaran, president of the LFP which represents the 42 professional clubs in the top two divisions.
"There is a strike called for the first two rounds of matches. We'll have to wait and see what happens overall. We'll try and reach an agreement and seek solutions to avoid the strike for the second round of matches."
Spokesman for the union, Luis Gil, told reporters: "We will meet tomorrow to try and narrow the differences. At least on this occasion they have put proposals on the table, something which didn't happen before. Now we need to evaluate them."