The Gunners have already made five new signings ahead of the start of the Premier League season next weekend, but are reportedly on the look out for a wide player to complete their squad.
Spanish news outlet Don Balon claim that Arsenal have marked Vazquez as their next transfer target and have submitted a €40m (£36.5m) offer to Los Blancos.
The 27-year-old winger struggled to nail down a regular place in Zinedine Zidane's Real Madrid starting line-up last season, making only 16 La Liga starts as the Spaniard competed alongside Cristiano Ronaldo, Gareth Bale and Isco for a spot in the attacking midfield.
However, Ronaldo's summer departure to Juventus and the arrival of former Spain national team manager Julen Lopetegui at Madrid means that Vazquez could have more opportunities to start matches at the Santiago Bernabeu next season, and so could be inclined to stay in Spain.
The Gunners need another man in the wings and could guarantee the 27-year-old more regular action. They have also been linked to Barcelona wide man Ousmane Dembele in recent weeks. Vazquez, however, is deemed the cheaper and more likely option for Unai Emery's side.
Vazquez has been at Madrid since 2007, coming through the Spanish giants' 'La Fabrica' academy to make his first team debut in 2015. He has since played 136 times for the Madrid senior side, winning one La Liga title in 2017, while also helping Zidane's side to three consecutive Champions League titles between 2016 and 2018.
The 27-year-old has also been capped nine times for his native Spain, making two appearances for La Roja during this summer's FIFA World Cup in Russia.