According to David Woods of the Daily Star, the Gunners are eyeing him as a replacement for manager Arsene Wenger when he steps down this summer having made a "tentative approach" for him last year, but Chelsea are said to be confident of landing him themselves with Antonio Conte expected to move on as well.
The Italian would be an ambitious target for Arsenal, and he'd arrive with an exceptional resume.
Allegri has won both Serie A and the Coppa Italia for the last three seasons running and could extend that to four this year—Juve are four points clear of Napoli in the former and have reached the final of the latter.
He also won the Scudetto with AC Milan in 2011—the Rossoneri's only title since 2004—and has contested two UEFA Champions League finals with Juve, albeit he lost to Barcelona in 2015 and Real Madrid in 2017.
Football writer Paolo Bandini praised him when he masterminded a comeback against Tottenham Hotspur in the Champions League earlier in the season and revealed his desire for him to take charge at the Emirates Stadium:
However, football commentator Adam Summerton does not believe Allegri is a realistic target for the Gunners:
Indeed, while he has played a large part in the Bianconeri's success over the last few years and deserves plenty of credit, so too has the club provided him with a strong platform on which to achieve it.
Arsenal have the potential to supply that, but after a decade of decline under Wenger, they have slipped away from the rivals, as demonstrated by their battle to simply finish sixth in the Premier League this season.
They also may not have Champions League football to offer either, should they fail to win the UEFA Europa League.
Rebuilding Arsenal will be a monumental task for whoever replaces Wenger. Allegri would be a fine choice, but with his stock so high, he may not wish to take on such a risky job.