The La Liga club have given PSG a list of six players that could be included in the deal, including Philippe Coutinho, Ousmane Dembele, Ivan Rakitic, Nelson Semedo, Malcom and Samuel Umtiti.
It is understood the French club want over £200m for the 27-year-old.
Sky Sports News reported on Tuesday that Neymar had told PSG he wanted to leave, while their manager Thomas Tuchel admitted on Wednesday that he knew the Brazilian was keen on an exit before the Copa America.
PSG sporting director Leonardo said last week that Neymar could leave the French champions "if there is an offer that suits everyone", while also confirming "superficial contact" with Barcelona.
According to Sky, PSG tried to convince Neymar to change his mind and the Brazil international did not tell his club where he would like to move to.
He was ruled out of the recent Copa America tournament - which Brazil won by beating Peru 3-1 in the final - because of an ankle injury sustained in a friendly against Qatar.
PSG signed Neymar for a world-record fee of £200m from Barcelona in August 2017.
Sky Germany's Max Bielefeld revealed on Thursday night on Sky Sports News that Neymar's agent is "really optimistic" the deal will go through, although any move is "very, very complicated".
"He wants to leave and Barcelona desperately want him back," he added.
Bielefeld also said that Neymar's father flew into London from Brazil with the agent on Wednesday for negotiations with representatives from PSG and Barcelona.
It is also understood that reports that Neymar's agent had been in talks with Juventus are not true.
"The only real solution in the current transfer window for Neymar is Barcelona," Bielefeld said.
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