Wilshere has not played since the summer due to an ankle injury, which required surgery in September.
A February return had been pencilled in, only for the 20-year-old to suffer a stress fracture in his right foot as he continued his rehabilitation.
With Arsenal languishing 17 points behind Premier League leaders Manchester City in fourth place, an earlier-than-expected comeback is sure to give the club a timely boost ahead of the season run-in.
However, a cautious Wenger says they will not rush the England player back, admitting the club are wary of him breaking down again during his recovery.
"Jack can be back within a month if all goes well," he said. "The recovery of his latest problems looks very good. The latest scan was very positive.
"We go cautiously step-by-step because we tried to push him back quicker in the first instance and he had a setback. So this time we have to be cautious and not go too quick. He can be back before then end of the season, of course."
While there was good news about Wilshere, Per Mertesacker's injury update was less positive. The German defender was carried off against Sunderland with an ankle injury and is predicted to be sidelined for a month.
Wenger said: "He had a reconstruction of his ankle in Germany before he joined us and the scan didn't look to positive yesterday [Monday].
"He went back to Germany to consult the specialist to see where we go from there and he is supposed to contact me this afternoon to tell me exactly what the specialist told him. Certainly in the next month he will not be with us."