Walcott could be the next high-profile player to leave the Gunners as he has less than a year left on his deal at the Emirates Stadium.
The Daily Mail reports that Juventus aim to take advantage of the situation and sign the winger at a cut-back price in January.
The Bianconeri sold off a number of wingers as Alessandro Conte decided to employ a 3-5-2 formation, but now the club appear keen to add the England international to their ranks.
Walcott has repeatedly rejected offers for a contract extension at the Emirates after falling behind Alex Oxlade-Chamberlain in the pecking order and his reluctance to commit his future has alerted top European clubs.
The 23-year-old started the first game of the season against Stoke, but was left on the bench in the next two league fixtures.