London's Sunday Express reported Allegri has verbally agreed to be Wenger's successor at the Gunners if the Frenchman turns down a two-year contract extension and decides to leave the club.
The Gunners board are anxious for an answer from Wenger, 67, and would like him to tell them about his future plans by the end of the month.
It is believed Allegri has been in discussions with Arsenal since last year after being identified as the London club's number one managerial target.
Questions surrounding Allegri's future have plagued the 49-year-old Italian this season and recent reports will only add further doubt to his future at Juventus.
Allegri, is set to leave Juventus after falling out with a number of senior players and has made no secret of his desire to work in the Premier League.
A shortlist of transfer targets has already been compiled by Allegri and passed on to the Arsenal board to green light if he is to take over the Gunners top job.
The list is said to include Lazio midfielder Sergej Milinkovic-Savic who was a target of Juventus last season.
