Rodriguez spotted arriving in Madrid
Photographs published in various Spanish media outlets on Sunday apparently showed the Monaco star at Madrid's Barajas Airport.
Rodriguez has been heavily linked with a switch to Real after lighting up the World Cup with a series of virtuoso displays for Colombia.
The playmaker, who turned 23 last week, won the Golden Boot at the finals in Brazil, with his haul of six goals leading his country to the quarter-finals.
Colombia were beaten 2-1 by hosts Brazil in the last eight, but Rodriguez's performances throughout the tournament announced his arrival on the global stage.
The former Porto man - who signed for Monaco for a reported €45 million last May - has made no secret of his desire to play for Real, telling Marca last week that the European champions are the "club of my dreams".
And it seems that Real president Florentino Perez is ready to grant him his wish, with the photos emerging over the weekend likely to increase speculation that Rodriguez will become the Spanish giants' latest 'Galactico' sooner rather than later.