Rodriguez is said to be available for a loan move, with his chances of first-team football at the La Liga giants limited and there are a number of suitors after his signature.
Everton and Villa will have to compete with Queens Park Rangers and Crystal Palace for the winger, while a number of Italian sides are also interested.
The player himself is said to be keen on playing Premier League football and it would seem a possibility that he could move to England before the end of the transfer window.
Rodriguez is one of Atleti's highest earners and they are looking to clear him off their books, considering his limited opportunities in the starting line-up at the Vicente Calderon.
The Uruguayan international has made 11 appearances for Atleti so far this season, but he has only started twice.