Szczesny is currently on loan at the Serie A side but talk of a permanent swoop has yet to be discussed.
Roma would like to sign the Poland international but are aware of his desire to make it as the number one at the Emirates.
And, according to Gazzetta dello Sport, the Italian club are lining up a move for Becker if they cannot tie down Szczesny on a permanent deal.
Becker has impressed for Internacional this season and made his debut for Brazil in October.
The 23-year-old would cost around £6m and Roma are set to hold preliminary talks in case they cannot complete a deal for Szczesny.