The Turin-based outfit are believed to have watching the speedy wide-man for some time and could offer him a Premier League escape route.
Liverpool and Manchester City are also interested, but Juve boss Antonio Conte has identified the 23-year-old as his main target ahead of the winter transfer window.
They are hoping to take advantage of the fact the England international’s current contract will expire at the end of the season.
He is said to be keen to stay in North London, so long as his employers put him on a salary which is equal to that of many other first-team squad members at the Emirates Stadium.
Walcott has already refused one offer from the club and is thought to be holding out for a salary of £100,000-per-week, some way offer of the figure the club are willing to pay.
The winger has struggled to hold down a position in Arsene Wenger’s starting XI this season mainly appearing from the substitutes bench.
Despite his young age, Walcott has made over 150 appearances in the red and white shirt since his arrival in 2006.
He is still seen as an important figure in the squad, but may be tempted to move on in order to reignite his career.