The Real Madrid winger has rejected a new contract offer from the LaLiga giants with his deal running out at the end of the season but the Gunners could face competition from Tottenham.
Vazquez, 29, has failed to hold down a regular starting place since making his first-team debut in 2015 with Cristiano Ronaldo, Gareth Bale and Eden Hazard in front of him.
And despite becoming a more integral part of Zinedine Zidane's team this season, with 19 league starts, he could be off.
It is understood he has rejected a new contract which would have seen him take a 10 per cent paycut and The Mirror claim the player's representatives have alerted Premier League rivals Arsenal and Tottenham about the availability of their client.
The Spain international has won two LaLiga titles and three Champions League trophies for Real Madrid.
Arsenal have worked often with Real Madrid in the transfer market over the years and currently have two players on loan in Martin Odegaard and Dani Ceballos.
Meanwhile, Tottenham boss Jose Mourinho could use his connections with his former club to lure Vazquez to the other side of north London.
Asked about his Real Madrid future last December, Vazquez, who can also operate as a right-back, said: "It's a difficult situation, we'll see what happens.
"For now, I'll keep working hard."
Arsenal could take advantage of their good relationship with Real Madrid by trying to sign Lucas Vazquez this summer.