Arsenal have been on the hunt for a new manager to take over from Wenger since his departure was announced last month.
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Allegri and Enrique have been heavily linked with the Emirates job to provide top-level experience in a time of change at the club.
But former players Mikel Arteta, Thierry Henry and Patrick Vieira are also understood to be under consideration as the club looks to a new era.
Chief executive Ivan Gazidis, head of football relations Raul Sanllehi and head of recruitment Sven Mislintat will ultimately decide on the new man.
And Sky Sports say Arsenal would like Allegri or Enrique to become the manager.
But Arsenal’s hopes of recruiting them is in doubt due to worries over the structure at the Emirates.
The new manager at Arsenal is not expected to wield the same level of power and influence Wenger exerted at the club.
Following his final home game as manager on Sunday, Wenger offered some advice to whoever takes over.
Wenger said: ”The new man inherits a team which is much better than people think it is, and I’m convinced, you can see the togetherness in the team is special.
“With the right additions this team will challenge next year.”
Former Arsenal defender Johan Djourou has backed Arteta to take over.
“That’s a difficult question because you’ve had such a long time with one coach that has been doing a great job for such a long time,” said Djourou.
“Of course, the last few years wasn’t so successful, but what he has brought to the mentality of the club… I think you cannot replace what he’s done.
“For me, it’s a new beginning, it’s a new era, it’s a new start. I think if you want to do a new start you have to go with someone who is maybe young, with a good philosophy, with a great mind as well. So, I don’t know, it’s hard to say.
“Allegri has good experience, Luis Enrique as well, but maybe I see someone like Mikel Arteta, maybe. Because he’s the second coach of Guardiola so he has this football mentality and he played for Arsenal before, so I don’t know.
“I think the future will tell but there are a lot of great minds and great coaches that could bring this team up in the future.”