Spanish publication 'OK Diario' reports that both Liverpool and Arsenal have registered their interest in the Spanish international forward.
The 26-year-old made 50 appearances for Real Madrid in all compeititons but made just 12 starts in La Liga and it is believed that he could be tempted by the offer of first team football.
Vazquez has been a regular fixture on the bench at the Bernabéu despite regularly proving his worth to Zinedine Zidane when called upon.
It is understood that the player isn't enamoured by the idea of another season spent on the bench and thus would be receptive to an offer from the Premier League.
Given his age and desire to prove himself it is believed that the Galician would find it very hard to turn down an offer of a regular starting berth with either side.
Zidane, for his part, is content with what the player brings to his team but could be willing to sanction a deal if it means he can push through a move for Monaco's Kylian Mbappe.
The Madrid boss admitted on Saturday that his side were in the market for a new striker following Alvaro Morata's move to Chelsea and it is believed that Monaco forward Mbappe is the club's first choice target.