Donny Van Der Beek was previously linked with a move to Premier League giants Manchester United.
The aforementioned report said that the two Spanish clubs will battle for the right to sign Donny Van Der Beek ahead of the resumption of the transfer market in the New Year.
Real Madrid manager Zinedine Zidane is keen to sign a new midfielder but compatriot Paul Pogba is seen as his first choice ahead of Donny Van Der Beek who could cost between €50m and €60m despite the fact that the Frenchman is still nursing an ankle injury.
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Tottenham Hotspur midfielder Christian Eriksen is also in the mix as Real Madrid attempt to sign a new midfielder.
Dutch international Donny Van Der Beek came into prominence in the Champions League last season as part of the Ajax Amsterdam team that surprised many by reaching the semifinals before they were eliminated by Tottenham Hotspur in dramatic fashion.
Donny Van Der Beek and Ajax Amsterdam eliminated clubs like Real Madrid and Juventus enroute to the last four of the competition.