The Poland international spent last season in the Italian capital and is thought to be closing in on another 12-month move to the Stadio Olimpico.
However, the deal is not yet done, which has left the door open for Ronald Koeman.
Reports claim that the Dutchman is eager to lure Szczesny to Merseyside and is willing to make him his No.1 following the exit of veteran stopper, Tim Howard.
Arsenal are unlikely to put up much of a fight, with Petr Cech and David Ospina stationed in north London.
As well as Szczesny, Everton are interested in Swansea’s Ashley Williams.
The Welshman, who shone for the Dragons at EURO 2016, could be a direct replacement for John Stones, who has been heavily aligned with a mega money move to Manchester City.