The 27-year-old has just one year left on his current deal and publicly expressed his desire to leave earlier this summer.
Real Madrid have been long-term admirers of Eriksen, who could be a cheaper alternative to £150million Manchester United ace Paul Pogba.
The Sun claim that the LaLiga giants will make a late move for the Dane with the Spanish transfer window remaining open until September 2.
Tottenham chairman Daniel Levy does not want to lose Eriksen on a free transfer next year and could cash in on the midfielder.
And Bernabeu chiefs believe Levy will be open to do business.