Unai Emery was sacked on Friday after his side threw away the lead to lose 2-1 against Eintracht Frankfurt in the Europa League.
Arsenal have now gone seven games in all competitions without a win.
Sky Sports News reports that Allegri has been contacted by the Gunners hierarchy as they begin their search for a permanent successor to the Spaniard.
Allegri departed the Turin club last summer after a hugely successful spell in which he led Juve to five Serie A titles, four Coppa Italias and two Champions League finals.
The outlet claims that the 52-year-old wants the job, although there are some doubts because of the standard of his English and his preference to take over in the summer rather than midway through a campaign.
Allegri matches the profile of coach that Arsenal are looking for, although his fellow Italian Carlo Ancelotti has also been linked with the position.
The experienced manager is currently in charge at Napoli, but has had a difficult season so far with the club struggling for form and his players falling out with the club hierarchy after refusing to go on a punitive training retreat.
Jose Mourinho was also being considered as a potential candidate for the role before he replaced Mauricio Pochettino at Tottenham Hotspur.
They are currently eighth in the standings, eight points outside of the Champions League places.
Thank you for reading 5 articles this month* Join now for unlimited access
Enjoy your first month for just £1 / $1 / €1
*Read 5 free articles per month without a subscription
Join now for unlimited access
Try first month for just £1 / $1 / €1
Get FourFourTwo Newsletter
The best features, fun and footballing quizzes, straight to your inbox every week.
Alasdair Mackenzie is a freelance journalist based in Rome, and a FourFourTwo contributor since 2015. When not pulling on the FFT shirt, he can be found at Reuters, The Times and the i. An Italophile since growing up on a diet of Football Italia on Channel 4, he now counts himself among thousands of fans sharing a passion for Ross County and Lazio.