If Rodriguez joins the Premier League side, it would see him reunite with manager Carlos Ancelotti for the third time, having played under the Italian at Bayern Munich and Real Madrid.
Everton have made an offer to Real for the player, which is now under consideration at the Bernabeu.
It is believed talks between the two clubs are progressing well and an agreement over a deal may be close.
Rodriguez has 12 months left on his current contract and is understood to be keen to make the move to Goodison Park on a three-year deal.
Last month, Ancelotti spoke about his admiration for the 29-year-old.
“James Rodriguez? I like, I like a lot as a player,” he said.
“I have to be honest, I like him a lot. But he is a Real Madrid player. I think he will still be a Real Madrid player.”