The Italians are reportedly keen on taking the Polish goalkeeper for a season as he faces up to a year on the bench with the Gunners.
The Poland international lost his place in the Gunners first-team last season to David Ospina, with the Colombian impressing after being a Premier League stage on which to perform.
He now faces further competition following the addition of Petr Cech this summer, and while Arsene Wenger would naturally like to have three keepers competing for a starting place, it is unlikely that the 25-year-old will be happy sitting on the bench.
That has led to Roma trying their luck, according to Sky Sports, with the report suggesting that early-stage negotiations are underway between the two clubs over a possible loan move.
Why a move to Roma makes sense
A loan move would make sense for Szczesny this season, as he is unlikely to be starting ahead of either Petr Cech or David Ospina.
He is likely only to feature in the cups, where he will also face competition for the goalkeeping spot, and that limited game time will stunt his development at an important stage in his career.
He has certainly benefitted from the regular first-team football he has enjoyed with the Gunners over the last few years, and a move to Italy would help to maintain the momentum that he has built.
In fact, a spell abroad could also aid his development, with the experience of playing in a new country and adapting to different playing styles, along with still turning out in the Champions League, likely to benefit him in the long term.