Last season, he made 38 appearances for the club, scoring six goals and assisting in another two.
Real Sociedad are looking to bring in the 32-year-old to their side ahead of next season as they look to improve upon their fortunes.
They finished just two points below the top half of the standings in the La Liga last season.
It is believed that Sociedad have made contact with the Bahia Group who represent Monreal over a possible transfer.
Monreal is understood to not be wanting to leave the Gunners at this point.
However, he does have one year remaining on his contract. This could work in Sociedad’s favor. The Gunners may be tempted if they receive a decent offer as it will eliminate the risk of losing the player for free during the January transfer window.