The move comes as the fallout continues from his record-smashing 222 million-euro transfer to Paris Saint-Germain.
This comes after Barcelona refused to pay a separate 26 million-euro bonus due to Neymar for signing a five-year contract renewal last year.
"In the lawsuit, the club demands the player return the already paid sum for his contract renewal as he has not completed his contract; 8.5 million euros in damages; and an additional 10 percent because of delayed payment," Barcelona said in a statement.
Neymar's family responded on Tuesday in a statement released by the company representing their interests.
"This news was received with surprise given that the player fully respected the contract valid at the time," said NN Consultoria.
"As for the bonuses owed for the signature of the contract in 2016, stipulated in the contract and not paid by FC Barcelona...(Neymar) has already started formal proceedings to recover the money before the competent court."
Although Neymar was under contract, Barcelona were powerless to stop his move to PSG earlier this month, because the French club met the buyout clause in his contract.
Barca added that the demand was filed against Neymar on August 11 with Barcelona's employment court, eight days after his shock move to the Parisian club.
The Catalan side have also lodged the claim with FIFA and the French Football Federation through the Spanish Football Federation, as they seek to take what they see as "the appropriate legal effects".
Furthermore, Barca urged PSG to pay the amount they are seeking if Neymar -- who is reported to earn 30 million euros a year after tax for the next five years in Paris -- cannot meet their demands.
PSG responded on Tuesday evening, announcing that they were "surprised" at Barcelona's move.
"Paris Saint-Germain reiterates that, like Neymar Jr., it has always respected all applicable laws and rules in its dealings and, once again, regrets the attitude of FC Barcelona," a club statement in English said.
Barcelona's latest act comes two days after Neymar strongly condemned the club's board after scoring a brace in a sterling home debut as PSG thrashed Toulouse 6-2 on Sunday.
"There are people in charge that shouldn't be there," Neymar told reporters.
"Barca deserve much more and everyone knows that."