Real Madrid will attempt to sign Kylian Mbappe on a pre-contract agreement in January, according to reports.
Los Blancos attempted to sign the France international this summer, but PSG refused to respond to their offers.
Madrid's third and final bid was worth £189m, but it was not enough to prise Mbappe away from the Parc des Princes.
Goal reports that the Spanish side will now focus their efforts on signing the 22-year-old on a free transfer next summer.
They will be permitted to begin talks with Mbappe from January 1, when the World Cup winner will be free to sign a pre-contract agreement with non-French clubs.
PSG have succeeded in keeping hold of one of their key players, and Mbappe will spend at least a season playing as part of a star-studded attack featuring Lionel Messi and Neymar.
As for Madrid, they completed the signing of Eduardo Camavinga before the transfer deadline at 11pm BST on Tuesday.
The 18-year-old midfielder was also a target for PSG and Manchester United, and had entered the final year of his Rennes contract.
It is mind-boggling that PSG felt able to reject a £189m bid for a player who can walk away for nothing in 10 months' time.
Their desperation to retain Mbappe for one more campaign might stem from their desire to win the Champions League.
An attacking trio of Mbappe, Messi and Neymar certainly gives PSG an excellent chance to lift the famous trophy for the first time.
More worryingly, it could be because money does not matter to the French club in the way it does to others.
There is no economic sense in rejecting almost £200m for Mbappe, but the Qatar-backed PSG clearly feel they do not need the cash.
A third possibility is that they believe Mbappe can still be persuaded to sign a new deal, but that looks like a long shot as things stand.
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