The La Liga champions have already had three bids rejected by PSG this summer, but they are poised to submit another in the coming days.
Barcelona are reportedly offering more money and more players after their latest proposal - an initial £36.6m loan deal with a compulsory purchase for £137m next summer - was turned down.
Real Madrid are divided over whether they should continue their pursuit of the Brazil international, with president Florentino Perez keen but manager Zinedine Zidane wary.
Meanwhile, Neymar is now open to the idea of remaining at the Parc des Princes for one more season.
There has been a softening of relations between the forward and PSG's sporting director Leonardo, and that could allow the Ligue 1 side to retain his services for another year.
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Greg Lea is a freelance football journalist who's filled in wherever FourFourTwo needs him since 2014. He became a Crystal Palace fan after watching a 1-0 loss to Port Vale in 1998, and once got on the scoresheet in a primary school game against Wilfried Zaha's Whitehorse Manor (an own goal in an 8-0 defeat).
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