The accolade is open to World Cup participants born on or after January 1 1993, from which Fifa's Technical Study Group will select a winner after the final.
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Varane started all but one of France's five tournament games in the centre of defence and showed signs of a return to his very best Real Madrid form from the 2012-13 campaign, though he was arguably at fault for Mats Hummels' winner in the quarter-final against Germany.
After beginning the World Cup as a substitute and entering the record books as Netherlands' youngest ever goalscorer against Australia, PSV star Depay cemented his place in Louis van Gaal's plans when he scored the second in the 2-0 win over Chile. Depay then became his country's youngest starter since 1938 against Costa Rica.
Lionel Messi, Neymar and Arjen Robben are among the nominees for the Golden Ball award for best player at the World Cup, while Manuel Neuer, Keylor Navas and Sergio Romero will battle for the Golden Glove prize.