The 28-year-old is available for around £30million and Spanish publication AS say a deal could be wrapped up as early as next week if the Gunners are willing to meet his valuation.
Having spent around £190m already this summer, Real are looking to recoup some cash through player sales while Gunners boss Unai Emery is desperate to bring in a new man to play out wide.
Vazquez has been on the fringes ever since making the step up to the senior Real Madrid team in 2015 and made only 15 starts in La Liga last season.
He has fallen even further down the pecking order since the arrival of Eden Hazard from Chelsea and Luka Jovic from Eintracht Frankfurt.
Despite falling out of favour, Vazquez would arrive at the Emirates with a wealth of experience having won a domestic title and three Champions Leagues in his time at the Bernabeu.
The reports from AS says the nine-cap Spain international is keen on the move but Arsenalcould also face competition from Paris Saint-Germain, Bayern Munich and Inter Milan.
Arenal might need to act quickly to avoid a bidding war as they are known to be working to an extremely tight budget this summer despite needing to strengthen in almost every area.