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According to reports in Catalan newspaper Mundo Deportivo, the experienced defender is on the radar of the Basque club ahead of the 2019-20 La Liga season.
The 33-year old is out of contract at the Emirates Stadium at the end of the 2019-20 campaign, and is likely to find his first team opportunities further limited following the arrival of Scotland international Kieran Tierney.
Real Sociedad shared left back duties last season between Theo Hernandez and Kevin Rodrigues, and with the latter going back to Real Madrid following his loan spell at the Anoeta - an on to AC Milan - Imanol Alguacil is short of cover.
Monreal's wages could be an issue, but Real Sociedad are hopeful that the promise of a greater first team role could persuade him to leave North London, after six and half season with Arsenal.
