The Scotland international is currently in the final few months of a season-long loan deal with the Basque side after being allowed to leave the Emirates Stadium last summer.
Although Tierney had hoped to gain regular playing time with Real Sociedad, he has ultimately seen his loan spell hampered by a couple of hamstring issues.
Tierney missed nine games in the first half of the season due to injury, before he was forced to miss around a month earlier this year due to another muscle problem.
As a result of his fitness issues, the left-back has had to settle for 14 starts in 21 competitive appearances for Imanol Alguacil's side.
Tierney's loan agreement did not contain a purchase option, so both clubs would have to negotiate a transfer fee if the Scotsman was to remain in San Sebastian.
However, according to journalist Charles Watts, Real Sociedad are unlikely to sign Tierney on a permanent deal in the summer transfer window.
The report claims that the La Liga club will struggle to match Arsenal's financial demands, and it is currently unclear whether they would want to sign Tierney even if they could agree a deal with the Gunners.
La Real are admirers of Tierney and have seemingly been impressed with his willingness to adapt to local life in Spain, but his recurring injury issues could deter them from even considering a permanent transfer.
Arsenal's preference was to sell Tierney last summer, but with the Gunners unable to find a viable suitor, they opted to send him on loan to Real Sociedad.
The North London club hoped the defender would prove his fitness and increase his market value ahead of the 2024-25 campaign.
However, Tierney's fitness struggles could make finding another permanent destination difficult if Arsenal are to hold out for a considerable transfer fee.
As it stands, Tierney will return to the Emirates Stadium this summer, with his Arsenal contract set to run until June 2026.