The two clubs are in discussions over a deal for Monreal, who has one year left on his contract in north London.
Monreal is likely to see his first-team chances decline at the Gunners with Sead Kolasinac and Kieran Tierney ahead of him in the pecking order once they both become available again.
Nacho Monreal may leave Arsenal to return to his native Spain and join Real Sociedad
The 33-year-old has made 250 appearances and scored 10 goals since joining the Gunners from Malaga in 2013.
He began his career at Osasuna, and has earned 22 international caps since his debut in 2009.
Monreal has started all three Premier League games this season, with Kolasinac largely absent over security fears after an incident involving himself and team-mate Mesut Ozil.
Tierney, a £25million summer signing from Celtic, is also awaiting his club debut as he works towards full fitness.
Monreal is thought to be keen on a return to his native country and would provide experience for the La Liga side.