Newcastle United are the latest club to have been linked with a move for the Colombian midfielder, while Manchester United, Arsenal and Everton are among the other Premier League outfits credited with an interest.
The 28-year-old's future appears to lie away from the Bernabeu, having failed to break his way back into Zinedine Zidane's first-team plans on a regular basis since returning from a two-year loan spell at Bayern Munich last summer.
Mundo Deportivo reports that Rodriguez has his heart set on a move to Atletico, who were also interested in his signature last year.
However, the Colombian, whose contract with Los Blancos expires next summer, would reportedly have to take a significant wage cut if he joins Diego Simeone's side.
Newcastle had been hoping to bring him in as a high-profile signing should their takeover be ratified by the Premier League, while Everton boss Carlo Ancelotti had been keen on a reunion having worked with Rodriguez at Bayern.
The 2014 World Cup Golden Boot winner has been limited to only 13 appearances across all competitions this season, the most recent of which came in October.