Juve coach Antonio Conte is a confessed admirer of the England international and is keen to add him to his squad.
The report claims that Juve have already made contact with Walcott's representatives and have been told that any move will be determined on whether or not Arsenal qualify for next season's Champions League.
When asked whether Walcott would remain at the Emirates Stadium for the long-term, his agent Colin Gordan responded: "We'll see. I don't know what will happen."
The 22-year-old joined Arsenal in 2006 from Southampton.