Wenger reacted to a glut of injuries in central midfield by signing the 31-year-old Swedish international from Spartak Moscow on the final day of the January transfer window on Friday.
However, reports soon emerged that Kallstrom had hurt his back playing beach volleyball and might not be able to feature for Arsenal for around six weeks.
Speaking after his side's 2-0 win over Crystal Palace in the English Premier League, Wenger played down the gravity of Kallstrom's injury but admitted that it would keep him sidelined until late February at the earliest.
"It's quite simple and we have to establish the truth," Wenger told Sky Sports. "He arrived here and fell on his back two days before.
"First of all, it's very difficult to identify a player of that stature who can come in on a free loan and then we made a medical (examination) of his back and we found out that he had a little micro-fracture.
"We have all played with micro-fractures, you know. When Spartak Moscow accepted to share the costs -- as long as he's injured, then they pay - I had a decision to make. We take nobody or Kallstrom, and it was Friday afternoon at five o'clock."
Asked when Kallstrom will be fit to play, Wenger replied: "I think earliest end of February, latest mid-March."