De Jong was strongly linked with a move to Barcelona during the summer transfer window, but Ajax ultimately managed to keep hold of the central midfielder.
De Jong's stock has risen further following his impressive performance for Netherlands in their 2-1 defeat to France in the UEFA Nations League at the weekend.
Tottenham Hotspur are believed to have had a bid in the region of £45m rejected by Ajax, meanwhile, as speculation surrounding the Dutchman's future gathers pace.
According to OK Diario, European champions Madrid are now firmly in the race for De Jong, who made 26 appearances for Ajax during the 2017-18 campaign.
De Jong moved to Ajax from Willem II in 2015, and represented Jong Ajax before stepping into the first team.