With Gunners boss Mikel Arteta experimenting with a new tactical setup in the wide areas this summer, Tierney had been left out in the cold in North London.
The Scotland international had already been demoted to second choice behind Oleksandr Zinchenko last season, but he has not even made a Premier League squad in the 2023-24 campaign so far.
Tierney did come off the bench for the final 14 minutes of the Community Shield last month, and Arteta recently insisted that the ex-Celtic man was still a part of his plans for the upcoming season.
However, even with Jurrien Timber out injured and Takehiro Tomiyasu suspended for Saturday's visit of Fulham, Arteta brought Jakub Kiwior into the starting lineup, and Tierney's name was nowhere to be seen on the teamsheet.
The 26-year-old has been training away from the main group of Arsenal players in anticipation of a summer exit, and a move to Newcastle United was strongly mooted several months ago.
However, the Magpies have since captured Lewis Hall on loan from Chelsea with an obligation to buy, forcing Tierney to look elsewhere in his search for regular first-team football.
Rather than continue in the Premier League, the left-back is set to try his hand at La Liga football with Real Sociedad, who can also offer him Champions League participation for the ongoing campaign.
According to Fabrizio Romano, Tierney landed in Spain this evening to finalise his switch to Imanol Alguacil's team, which will be a straight loan with no option or obligation to buy.
The former Celtic man remains under contract in North London until 2026 and has won over the Arsenal
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