SunSport also reported how Barcelona's vice-president Jordi Cardoner claims the Brazilian is desperate for a second stint at the Nou Camp.
Barcelona scout Andre Cury, who brokered Neymar's move from the Blaugrana to Paris Saint-Germain, was said to be involved in the talks, too.
According to the French publication, the divorce between Neymar and PSG is a reality and they have left the possibility open that he may not start pre-season with the Parisian side.
La Liga president Javier Tebas has said he would rather Neymar does not return to La Liga if he can not improve his behaviour.
"On the other hand though, the club members, who vote in the presidential elections, are not quite as keen".
It would be a win-win situation for both clubs as it would pave the way for Neymar to make his much-wanted return to Spain.More news: Newark Airport resumes flights after brief closure due to emergency
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Barcelona are eyeing a potential homecoming for Neymar two years after receiving a world-record £198m fee for the Brazilian.
Barcelona set to announce the signing of the 2018 FIFA World Cup victor Antoine Griezmann after his release clause dropped to €120 million on Monday. He will now go to Belo Horizonte to watch the Copa America Semi Finals between Brazil and Argentina.
Neymar looks to be inching closer to a return to Barcelona.
Tebas also questioned Neymar's behaviour on Sunday, saying he did not want Barca to sign the superstar forward again.
Sport believe that the only way Barcelona could pull off a deal for Neymar would be in a cash plus player deal which is where Coutinho comes into the fold.