Barton, 29, was a surprise name in the Newcastle line-up for Saturday's game at home to Arsenal after a summer in which he had criticised his club's owners on Twitter and been made available on a free transfer.
With a long list of controversies overshadowing his undoubted talent, it is reported there have been few serious enquiries for his services since he was placed on the transfer list. In the aftermath of the announcement that he was available for transfer, Barton tweeted a request for a phone number for the master bridge builder Isambard Kingdom Brunel.
Having been given a chance for Newcastle's season opener, Barton was in the spotlight again. During the 0-0 draw, he was stamped on by Song and then confronted Gervinho over a perceived dive before going down theatrically after receiving a slap to the face. Gervinho, making his debut, was sent off.
Wenger had said he felt Barton should have received the same punishment as Gervinho in the immediate aftermath of the incident, but it appears there is no lingering ill feeling.
"I like the courage of the player," Wenger said in The Guardian. "Nobody dismisses signing Joey Barton but I don't think we envisage doing that at the moment. He's a good player. Sometimes there's trouble with his temper but he's a great player."
He joked: "Maybe the way to sort Barton out is to sign him. Maybe that's the solution."
Barton has said he has seen the Gervinho incident again and, while he accepts there was contact, he still believes there was an attempt to unfairly win a penalty.
"I have to take back blatantly calling him a cheat but he was still looking for a penalty and thankfully the referee knew what he was up to," he told Radio 5 Live. "I have to take responsibility for my actions and I should not have gone over and lifted him by his shirt. That was not the correct thing to do.
"We do push the boundaries but I felt disappointed by Gervinho trying to win a penalty. I am not a fool, I realise there is slight contact, but in my opinion he was seeking to gain an unfair advantage. Sometimes you react and I am not proud of the way I reacted.
"I did go down easily but it is not okay to hit me because it's me. I have been hit harder at school. It does not make what he did right or wrong but he raised his hand and the rules are you can't do that. I have made a meal of it but I knew Song had stamped on me and I was slightly aggrieved by that because I am aware the spotlight is on me at certain times, because of my stupidity in the past and people don't condone me easily.
"In 20 years of the Premier League there has been an influx of diving, and I sound hypocritical because I am guilty of going down too easily, but if I didn't and I stayed on my feet, I could be making my own team suffer if the officials had not seen it."