"He has been practising with us," Wenger said on the club's website on Tuesday. "Will I sign him again in January? I don't know. I don't rule it out. He is sharp."
Arsenal's 228-goal record scorer, now 35, returned to the club last January, netting against Leeds United in the FA Cup and grabbing a late winner at Sunderland in the Premier League.
Wenger was delighted with Henry's impact and believes he could be a valuable asset if he came back for a third spell, especially if the club lose Ivory Coast striker Gervinho and Morocco forward Marouane Chamakh to the African Nations Cup.
"Last year I did it because we lost Gervinho. It was a good connection. This year we lose Gervinho again as they are playing in the Africa Cup of Nations two years in a row," said Wenger whose team have suffered an indifferent start to the season.
"So we will be confronted with a shortage. Particularly if Chamakh should go. I cannot stand in his way. Then we will be short," added the Frenchman ahead of Arsenal's Champions League match at home to Montpellier on Wednesday.
Wenger knows Henry, who has finished the season in the United States after his Red Bulls team lost to D.C. United in Major League Soccer's Eastern Conference semi-finals earlier this month, is influential both on and off the pitch.
"You know what he gives you. He gives you hope, especially when he comes on. That is the most important thing," he said.
"He is a communicator. An extrovert. Very intelligent. He can only give good advice to players because he was in their position when he arrived here."