The Gunners lost Kevin Betsy, who coached the U23s and Dan Micciche, who was in charge of the U18s earlier in June - Betsy and Micciche taking the manager and assistant manager's roles respectively at League Two side Crawley Town.
The club have begun their recruitment process and the Evening Standard reports Wilshere recently spoke with academy manager Per Mertesacker about the roles.
The report adds that Arsenal are also considering internal candidates but Wilshere is held in high regards by a number of the current staff.
Wilshere is currently without a club having left Danish side Aarhus in May. The 30-year-old is hopeful of continuing his playing career but the prospect of coaching appeals to him too.
"I didn't ever reach my full potential as a player — and everyone knows that," Wilshere said in May.
"I don't think I will ever accept it until I've reached that as a coach. It's a difficult thing to accept and that's why I still have a big, burning desire in me to be successful in the game.
"I just feel like there's something still in me that wants to get to a high level. And that probably won't ever be as a player again. So if I can do that as a coach, why not?"
Arsenal is a big part of Wilshere's life. He came through their academy before excelling in their first team.
Wilshere made a return to the club in the 2021/22 season, when he was looking for a new side, training with them to maintain his fitness.
The midfielder also worked with the club's U23 and U18 players and spoke of how much he enjoyed the experience.
"I think I would love to go back to Arsenal and coach, once I have fully made that decision," Wilshere said in May when asked about retirement.
"That decision might be in a week. It might be in two years. If I don't get a club to play for, before pre-season or halfway through pre-season, I am not going to go to Arsenal to keep my fitness up, because then I would feel that I am holding onto something that is not there. With a coaching role, maybe. We will see.
"When I went back to Arsenal I was coaching every day with the under-18s and 23s, I loved every minute of it. I really enjoyed being back on the pitch and coaching."
The Standard's report adds Wilshere has a 'very good relationship' with Mertesacker, who he played alongside during his time as an Arsenal player.