Rodriguez, who was a summer target for Sunderland, has only been given fleeting chances to impress at Atletico this season and the feeling is he may be available for transfer in the new year.
A skillful dribbler, he looks a player who could add quality and experience to the R's squad as they battle against the drop.
Harry Redknapp says his side have no money to spend on permanent deals, but they are prepared to bring in some short-term options to improve the team.
The R's are battling against the drop and they may feel that Rodriguez's quality from wide areas could help them in front of goal.
The London club have only managed to score 17 times in their 16 Premier League matches to date and Redknapp is known to be targeting attacking additions at the next available opportunity.
QPR may be hopeful that Rodriguez is a viable option because of Atleti's financial struggles. The La Liga side are heavily in debt and need to clear some of their fringe players out to reduce the wage bill.