Campbell has been training with his former side in recent months and told Soccernet this week that he hoped to complete a deal soon after some "technicalities" in the deal were resolved.
It had been suggested that Wenger would use the reserve match against West Ham to gauge whether the 35-year-old was still capable of playing at the highest level and it appears the defender did enough to convince the manager he was worthy of a deal until the end of the season.
"I think he will sign later today," Wenger said on Friday. "There are no major problems and it's all sorted out now.
"It gives us an opportunity to have one more centre-back. At the moment, (Philippe) Senderos has been injured and because we haven't extended his contract we want to give Senderos the opportunity as well to join another club."
Asked what Campbell would bring to the side, he said: "Experience, good advice and, physically, he has worked very hard. We gave him the opportunity to relaunch his career. For him, it is a good opportunity to show he can still play in the Premier League."
Wenger suggested Campbell could force his way into the first-team if his performances warranted it and added: "I think he's shown he was dedicated, motivated and very happy to be back. Many, many players would be happy to be back at Arsenal."
Meanwhile, Arsenal youngster Mark Randall has joined League One side Milton Keynes Dons on loan until the end of the season.