Los Blancos have made a €55 million (£50m/$61m) offer for the 22-year-old but Ajax are demanding €60m (£55m/$67m) for his services. Van de Beek has been on Real's radar since last season and they have finally decided to act on their interest due to a lack of depth in midfield heading into the 2019-20 campaign.
Paul Pogba has also been a long-term target of the club, but signing the France midfielder is proving difficult and a deal would have to be completed before the English transfer window closes next week. Van de Beek is seen as a more plausible transfer and Real believe that he could also add goals from midfield as they look to recover from a disappointing 2018-19 season.
The Netherlands international confirmed that Real were interested in securing his services, but played down talk that he was a backup plan for Pogba at Madrid.
"Of course I hear stuff like that from time to time and it's always good to hear about it," he told Fox Sports.
"But the situation is still the same, even though the media keep creating new stories. Of course there is interest, but people tend to rush to conclusions about that."
The arrival of Van de Beek at Santiago Bernabeu will also help James Rodriguez move away from the club. Real Madrid are eager to bring some money back into the coffers following an expensive summer, where they have already signed Eden Hazard, Luka Jovic, Eder Militao, Ferland Mendy and Rodrygo for over €300m in total.
Gareth Bale is another option to leave this summer, but the club has not found a suitor following his collapsed move to China. That leaves James the most likely departure, with Atletico Madrid eager to sign him on a permanent transfer. Napoli are also interested, but only on an initial loan deal.
Real are hoping to get more than €42m (£38m/$47m) for the Colombian and thus would be more likely to sell to local rivals Atletico than loan him out to Napoli.