The Dutch giants had also set their sights on the promising Vitesse midfielder, but he is expected to make the move to the Europa League winners instead
Sporting director Marc Overmars has admitted that the Eredivisie champions cannot compete financially with Premier League giants Chelsea.
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"Ajax were very interested in Van Ginkel and we were willing to go quite far," Overmars was quoted as saying by Voetbal International.
"But if we have to compete with one of the big clubs from England it is always going to be difficult for us. We cannot beat Chelsea's financial resources."
Van Ginkel has a contract with Vitesse until the summer of 2015, but the Arnhem side are ready to cash in on the 20-year-old for a fee in the region of €10 million (£8.4m).
The midfielder has been one of the stars of the Under-21 European championship in Israel this summer as Netherlands have qualified for the semi-finals after wins over Germany and Russia.