Szczesny has spent the last two seasons on loan at Roma after falling behind Petr Cech and David Ospina in the pecking order at Arsenal.
And the Polish keeper appears to have impressed Serie A rivals Napoli during his time at the Stadio Olimpico, where he made 81 appearances.
According to Italian outlet Il Mattino, Szczesny has agreed to join Napoli this summer with other reports suggesting a three-year deal is on the table.
The 27-year-old's representatives are said to have met with Napoli officials earlier this month.
Arsenal Transfer News: Wojciech Szczesny has reportedly agreed Napoli deal
Wojciech Szczesny fell behind David Ospina and Petr Cech at Arsenal
Arsenal goalkeeper Wojciech Szczesny agrees Napoli transfer
Napoli are reportedly open to letting first-choice keeper Pepe Reina leave, with the Spaniard being linked with a return to England at Newcastle.
Szczesny helped Roma to a second-place finish in Serie A this season, keeping 14 clean sheets in his 38 league outings.
The Poland No 1 admitted earlier this month that he is yet to make a decision on his future, suggesting he could remain at Roma and attempt to kick-start his Arsenal career.
Wojciech Szczesny has enjoyed two successful seasons at Roma Petr Cech rushes to David Ospina after FA Cup final win Sun, May 28, 2017
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Petr Cech and David Ospina celebrate Arsenal's FA Cup win
He told BT Sport: “Staying at Roma? I’m happy here. I feel I’ve grown as a goalkeeper but for me the time to make a decision will come in the summer.
“For now I’m an Arsenal player, as soon as my loan finishes, I have to come back [to London]. I’ve always said I’d love to be back there.
“Like I said, what will happen definitely won’t be decided now but in a month or two months’ time.”