Vazquez has made regular appearances for Real under coach Zinedine Zidane, but the 26-year-old remains with just 34 starts in all competitions since breaking into the first team back in 2015.
The Spanish international has already featured on four occasions at the start of the new campaign but according to Don Balon, both Arsenal and Liverpool are ready to offer Vazquez more game time in the Premier League.
It has been suggested that Arsene Wenger and Jurgen Klopp are admirers of Vazquez, and with both managers having funds to spend before the end of the summer window, an official approach could be imminent.
Vazquez has spent all of his career in his homeland, with a year's loan spell being spent at Espanyol during the 2014-15 campaign before the wideman became part of Real's star-studded squad ahead of the following season.
A total of eight goals have been scored in 87 appearances by Vazquez, who still has four years remaining on his contract.