The England international is widely expected to leave this summer after losing his place to David Raya this season.
Ramsdale featured in 72 Premier League matches across his first two seasons as an Arsenal player, but he has had to settle for just six top-flight appearances during the current campaign.
After playing in Arsenal's opening four Premier League games, Ramsdale's last two league appearances took place against Brentford, with Raya ineligible to face his parent club.
Having been forced to settle for the number two spot, the 25-year-old is set to depart to gain regular playing time elsewhere.
Arsenal considering Szczesny reunion
Arsenal are expected to trigger their buy option for Raya, but they would still need to recruit another goalkeeper to replace Ramsdale.
According to The Standard, Arsenal are considering a move to bring Szczesny back to the Emirates Stadium ahead of the 2024-25 campaign.
Juventus are believed to be willing to offload Szczesny this summer, having already identified Monza's Michele Di Gregorio as his potential replacement.
With only one year remaining on his contract, the Poland international represents a relatively cheap option for the Gunners.
As a product of Arsenal's academy, the 34-year-old would be listed as a homegrown player, which would help the Gunners meet the Premier League's Profit and Sustainability Rules.
Arsenal boss Mikel Arteta will be fully aware of Szczesny's capabilities after he spent five years playing alongside the Polish goalkeeper, before the Spaniard hung up his boots in 2016.
Steele, Ramaj feature on Arsenal shortlist
However, there may be some doubts as to whether Szczesny would be willing to be Raya's understudy after making 250 appearances during his five-year association with Juventus.
In addition to Szczesny, Arsenal have also added Brighton & Hove Albion's Jason Steele to their list of potential summer targets.
The shot-stopper has made 16 Premier League appearances this term, having spent the campaign competing with Bart Verbruggen for Brighton's number one shirt.
Ajax goalkeeper Diant Ramaj is also believed to be on Arsenal's radar after impressing during a difficult season for the Dutch giants.
At the age of 22, the German represents a different option to the experience of Szczesny and Steele, but he may be reluctant to give up a regular starting place at the Amsterdam Arena.