The Spain international has reportedly emerged as a leading target for Arsenal head coach Unai Emery, who succeeded Arsene Wenger in May.
According to MARCA, Emery is ‘in love’ with Vazquez’s direct running style on the wing.
But the Spanish publication - renowned for it’s close ties to Real Madrid - claim Liverpool and Chelsea are also in the hunt.
Liverpool are reportedly eyeing Vazquez in addition to his Los Blancos team-mate Marco Asensio.
While Chelsea are said to be closely following any developments regarding the 26-year-old’s future.
Vazquez featured 53 times in all competitions last term but only 27 of those appearances came from the start.
He chipped in with eight goals and 16 assists last term and forced his way into new Real Madrid boss Julen Lopetegui’s World Cup squad.
Lopetegui was sensationally sacked by Spain on the eve of the tournament after his deal with Los Blancos came to light and replaced by Fernando Hierro.
Vazquez has made two appearances at the World Cup for Spain, who qualified from Group B to the last 16.
MARCA claim Vazquez - also attracting interest from AC Milan - is solely focused on the World Cup.
He is reportedly happy at the Bernbaeu and is under contract with Real Madrid until the summer of 2021.
Reports in Spain earlier this week suggested Arsenal have offered Vazquez a triple-your-money deal worth £7m (€8m) per season to move to the Emirates Stadium.sources  
MARCA.com