The Iceland international was brought in to replace the departing Emiliano Martinez in 2020 and acted as backup to Bernd Leno during the 2020-21 season.
Runarsson - who arrived from Dijon for a mere £1.8m - was restricted to just six appearances in all competitions that year and did not start a single Premier League game for the Gunners.
The 27-year-old's only outing in the top flight came as a substitute in a 2-1 loss to Wolverhampton Wanderers after Leno was sent off, while he also started one EFL Cup game and four Europa League group-stage matches.
Following the arrival of Aaron Ramsdale, Runarsson was sent out on loan to Belgian top-flight side OH Leuven for the 2021-22 season and kept two clean sheets in 20 appearances in all tournaments.
Runarsson has since been put through his paces at London Colney in pre-season and travelled with the Gunners to the USA earlier this summer, but he does not appear to have a future at the club.
According to the Evening Standard, Antalyaspor hold an interest in signing the former Reykjavik and Nordsjaelland man on loan and are now in discussions with Arsenal over a deal.
Runarsson's contract with the Gunners still has another two years left to run, and the North London club would likely prefer to sanction a permanent exit given the competition for spots in between the sticks.
Matt Turner has arrived from New England Revolution to cover for Ramsdale this season, while 20-year-old Estonia international Karl Hein will also be aiming to make the step-up to senior club football.
Arsenal allowed Leno to depart for Fulham in a deal that could eventually reach £8m, while Arthur Okonkwo will gain experience out on loan at Crewe Alexandra.
Having been busy in terms of incomings earlier this summer, Mikel Arteta's side are now expected to axe numerous fringe players before the end of the transfer window.
As well as Leno, Matteo Guendouzi, Alexandre Lacazette and Dinos Mavropanos have already left the club, while Lucas Torreira's £5.5m move to Galatasaray was confirmed on Monday.
Out-of-favour centre-back Pablo Mari is understood to be pining for a return to Italy after doing well on loan at Udinese, with Monza and Hellas Verona battling to sign him.
Hector Bellerin, Ainsley Maitland-Niles and Nicolas Pepe's futures are also shrouded in uncertainty, but the latter - who cost the Gunners a record £72m - is supposedly failing to attract serious interest.
Arsenal will aim to continue their 100% start to the new Premier League season when they tackle Leicester City at the Emirates Stadium on Saturday afternoon.