The 28-year-old was centre of attention when the two sides drew 0-0 at St James' Park on Saturday as he was involved in the incident which saw Gunners debutant Gervinho sent off.
That came after a summer which saw Barton made available on a free transfer - and linked with Arsenal in some quarters - following his criticism of the Magpies' transfer policy.
In comments quoted in The Guardian, Wenger said: "Nobody dismisses signing Joey Barton but I don't think we envisage doing that at the moment.
"I like the courage of the player. He's a good player. Sometimes there's trouble with his temper but he's a great player."
He added jokingly: "Maybe the way to sort Barton out is to sign him. Maybe that's the solution."