The Iceland international was signed by the North London club in 2020 following the departure of Emiliano Martinez to Aston Villa and acted as Bernd Leno's number two in his debut season.
Runarsson's prospects of Premier League game time slimmed further when Aaron Ramsdale was brought to the club, and he was loaned out to Belgian side OH Leuven last year.
Runarsson kept two clean sheets in 20 games for Leuven in all tournaments and was involved in Arsenal's pre-season amid persistent links with an exit.
Arsenal have not been able to shift the 27-year-old permanently, but the Gunners have now announced that he will spend the next year on loan at Alanyaspor.
Runarsson undertook his medical and signed his contract with the Turkish club earlier on Monday, and he is due to return to Arsenal for the 2023-24 season.
The former Dijon man is only under contract at the Emirates until the summer of 2024, meaning that the Gunners will likely try to offload him for good next year.
Runarsson has made a mere six appearances for Mikel Arteta's side since arriving from Dijon for £1.8m in 2020, one of which came in the Premier League as a substitute against Wolverhampton Wanderers.
The Icelandic shot-stopper has also kept two clean sheets in four Europa League outings for Arsenal and has featured once for the Gunners in EFL Cup action.
Runarsson is not the only Arsenal goalkeeper to have been loaned out in the current transfer window, with Arthur Okonkwo joining Crewe Alexandra earlier this summer.
Meanwhile, Leno left permanently for Fulham after the arrival of Matt Turner, but Marek Rodak has kept the Germany international out of the Cottagers' XI for the time being.
In terms of incomings, Arsenal may have already completed their summer business, although Arteta has affirmed that his side are still open to welcoming new faces to the club if the deal is right.
While Leicester City's Youri Tielemans is a long-term target, Arsenal are supposedly prepared to wait until 2023 before attempting to sign the Belgian on a free transfer.
A move for Villarreal's Yeremi Pino has also been mooted, but it is understood that technical director Edu's recent flight to Spain was not related to links with the 19-year-old.
Arsenal will aim to begin their Premier League season with three consecutive wins when they travel to newly-promoted Bournemouth on Saturday evening.