Per Graeme Bryce at The Sun, Massimiliano Allegri's men view the Pole as the perfect long-term replacement for Gianluigi Buffon.
Szczesny has spent the last two seasons on loan at Roma, and he has just a year left on his current Arsenal deal, but the Gunners are demanding at least £13 million for his services.
According to the report, Szczesny is keen to remain in Italy, but Roma want him off the wage bill, and although Napoli have shown an interest, they also cannot afford Arsenal's asking price.
Szczesny enjoyed a fine season last time out, helping Roma to finish second in Serie A, and he also ended the season with more clean sheets than any other goalkeeper in Italy's top flight:
Buffon had also appeared to hint that Szczesny would be his replacement in Turin, speaking ahead of Italy's World Cup qualifier against Liechtenstein, earlier this month.
"I am ready to retire with Donnarumma here and Szczesny at Juventus. I know I won’t last forever," he said, according to Calciomercato.com.
Szczesny is due to return to Arsenal when his Roma loan expires this summer, and if he stays, will rival Petr Cech for the No.1 spot, with David Ospina and Emiliano Martinez as back-up.
Goal's Chris Wheatley managed to compared the records of Cech and Szczesny last season, and statistically, the Pole has the edge:
Szczesny's return could, therefore, be well-timed with Cech having turned 35 last month, and reaching the end of a long and hugely successful career.