In a statement on their website, the La Liga club noted the left-back has agreed a two-year contract and will be presented on Monday.
The transfer brings an end to a six-year spell at the Emirates Stadium for the defender, who joined Arsenal from Malaga in 2013.
The Gunners sent their best wishes to the left-back on Twitter:
"We would like to thank Nacho for his contribution to the club, and everyone at Arsenal wishes him the best for his future," said Arsenal in their own statement.
Sociedad also posted the following clip on their Twitter account:
Monreal's Arsenal career didn't get off to the best of starts, as he initially struggled to cope with the pace and physicality of English football.
However, he was able to make strides at the Emirates Stadium and eventually established himself as one of the most dependable left-backs in the division.
The Spaniard was strong in his defensive duties and would rarely be caught out of position. Monreal has never been the most dynamic of attackers, but when he ventured forward, he would use the ball well and knit play together intelligently on his flank.
Now Monreal is in the twilight of his career, and the signing of Kieran Tierney from Celtic in the summer has pushed him down the pecking order at Arsenal.
Dan Critchlow of the Daily Cannon praised the veteran for the way in which he has conducted himself amid speculation and has called on the club's other left-backs to step up:
Dan Critchlow @afcDW
Pressure is on Kolašinac and Tierney’s return now then. Probably the right time for Monreal to move on but the next few weeks are going to be interesting. As for Nacho himself, handled himself really well throughout this saga, kept his head down, got his move without any drama. https://t.co/kLzJwCZyhY
The defender will be remembered fondly for his efforts at Arsenal and for the part he played in a number of successes—Monreal won three FA Cups with the Gunners.
Real Sociedad supporters will welcome the arrival of a footballer who has so much experience to call on. The announcement of the signing will also be a boost a day on from their 2-0 loss to Basque rivals Athletic Bilbao on Friday.