Ozil has been linked with a move away from the Emirates next summer as contract talks with the club remain at an impasse.
As first reported by The Independent, Manchester United are looking to touch base with Ozil in January once the German becomes free to talk to other clubs.
And speaking to BBC Radio 5 Live, Keown said: “I think he's testing [Arsene] Wenger more than any other player has. I think it was Belarus the other week, he didn't fancy going. He didn't want to go.
“Suddenly he was injured again, he played nine minutes against West Bromwich Albion on the Monday, so how did he get injured there?
“I think in some departments he's already left. Psychologically, mentally, he's already left the football club. Maybe Wenger is now trying to do the best deal he can to get some compensation for him.”
Ozil has struggled for game-time this season, making just three starts for Wenger’s side out of their opening seven matches.
Injury has played a factor in the 28-year-old’s absence though this has failed to end speculation that the midfielder has fallen out of favour under Wenger.
“I think [Wenger] doesn't want that sort of player around the youngsters in the group,” Keown continued.
“They played exceptionally well at Chelsea, where [Alex] Iwobi did a really manful job – I know he hasn't got the talent of Ozil, of course, but suddenly Arsenal could compete in both departments defensively and attacking-wise. It was probably the best performance I've seen in a couple of years.
“I was really impressed and it seemed like every player was giving their best. Sanchez is different. He loves playing football, just give him a ball and he'll play and that's all he wants to do. But Ozil, I think he's downed tools, he wants to go and it's going to get messy.”
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