The German is out of contract at the end of the season and has been linked with a surprise reunion with Jose Mourinho at Manchester United next term.
In an interview with 'BBC Radio 5 Live' Keown suggested that the player simply does not want to play for Arsenal any longer, saying that the reason he was left out of the squad for the trip to BATE Borisov in the Europa League was because the player had no desire to travel.
'I think he's testing 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,' said the former England international.
'Suddenly he was injured again, he played nine minutes against West Bromwich Albion on the Monday, so how did he get injured there?'
Keown, a former member of the Arsenal 'invincibles' side says that Ozil is no longer interested in playing for Arsenal, with his mind already very much elsewhere.
'I think in some departments he's already left,' he explained. '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.'
'I think [Wenger] doesn't want that sort of player around the youngsters in the group. 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.'
Ozil has made just five appearances so far this seson and has not played ninety minutes for the 'Gunners' since the win over Bournemouth almost a month ago as he slides slowly nearer to the exit door.