Zinedine Zidane's side are competing with Barcelona for the Brazil international, who is keen to leave the Parc des Princes before the European transfer window closes on September 2.
PSG rejected an offer from Barcelona earlier this week, and the La Liga champions are struggling to meet the French club's demands.
Madrid now spot an opportunity to jump to the front of the queue, with Juni Calafat, the club's head of international football, currently in Paris in an attempt to finalise the transfer.
Los Blancos are able to offer the type of players-plus-cash deal that PSG are thought to be interested in.
And Madrid hope to be in a position to unveil Neymar at the Santiago Bernabeu next week.
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