Wijnaldum, 30, is set to leave Liverpool at the end of the month when his contract expires at Anfield.
A number of Europe's biggest clubs have shown an interest in the midfielder, who is preparing for the European Chanpionship with the Netherlands.
His agents met with Barca president Joan Laporta twice last week to discuss a three-year deal at Camp Nou.
Ronald Koeman first wanted to sign Wijnaldum last summer and, after Laporta confirmed the Dutch coach would remain in charge next season, he finally looked set to land his compatriot.
However, sources close to the negotiations have told ESPN that PSG have made a much more generous financial offer to Wijnaldum, who at this moment is leaning towards a move to Paris.
Sources at Barca, meanwhile, acknowledge that there is "huge" competition for the player and expressed doubts about completing the singing.
Bayern Munich have also shown an interest in Wijnaldum.
Juventus and Inter Milan have also held talks with Depay.
The Blaugrana have also activated a clause to recall right-back Emerson Royal from Real Betis for €9 million.