Gunners boss Unai Emery is considering Vasquez as an alternative to Wilfried Zaha.
Crystal Palace rejected Arsenal’s opening £40m offer for their star man last week.
And Vazquez has since emerged as Emery’s latest summer transfer target.
It follows failed moves for Bournemouth’s Ryan Fraser and Yannick Carrasco of Dalian Yifang.
Vazquez played 47 times for Los Blancos last season, mainly on the right wing.
The 28-year-old initially wanted to stay and fight for his place under Zinedine Zidane next term.
But with the arrivals of Luka Jovic and Rodrygo he is likely to find his playing time restricted.
Despite not being able to provide Champions League football, Arsenal’s offer is seemingly proving tempting to Vazquez.
Spanish publication ABC.es say he is giving serious consideration to the option of moving to England.
Starsport understands that Paris St-Germain, Inter Milan and Bayern Munich have all asked about the Spain international.
But his preference is believed to be to move to the Premier League.
Meanwhile, The Times say that Kieran Tierney looks certain to leave Celtic this summer if Arsenal bring a £23m offer to the table.
The Bhoys have already signed Belgian left-back Boli Bolingoli from Rapid Vienna for £3m.
Arsenal now just need to beat Napoli to the capture of Tierney.
The Gunners are also in the market for a central defender and there is growing confidence they will win the race to sign Saint-Etienne’s William Saliba.