Neymar is thought to have made it clear to PSG numerous times this summer that he wanted to return to La Liga, having left Spain in 2017.
The Parisian club paid £205million for Neymar, smashing the transfer fee world record.
But the Brazilian has only managed to obtain domestic trophies in his time with PSG, having been knocked out of the Champions League at the last 16 stage two years in a row.
And with the European transfer window nearing its closure, Balague has revealed that PSG were demanding a final offer be made for Neymar by both Barca and Real earlier today.
He tweeted: “Barcelona AND Real Madrid have been asked to put a final offer before 12pm on Friday. Juventus also in the equation.
“The latest twist: in the last few days the relationship between #Neymar, his father/representative and #PSG has improved dramatically (in a VERY positive way).”
Balague also laid out the terms that PSG would consider listening to for Neymar’s signature – with both Gareth Bale and Luka Jovic, who Real only signed this summer, on the table.
“Real willing to offer Bale (but he doesn’t want to go) and Jovic, but PSG only want Vinicius (and Real don’t want to let him go).” he added.
“Real saying they are not talking to PSG… But they are big time!”
Neymar spent four seasons at the Nou Camp with Barca and won a Champions League with the Catalan club in 2015.
But Balague says the 27-year-old’s ex-club are worried that Neymar is slowly slipping out of their grasp.
“Barcelona pushing hard as they feel Neymar getting away,” he added.
Neymar is also said to be happy to reduce his wages significantly in order to push through a move that would see him return to Spain.
“The player is happy to reduce his wages from 38 million euros net to 23 million euros net,” Balague told BBC Radio 5 Live last week.
“It is an interesting situation at Barcelona. [Barca] feel they have to try really hard to get him back.”