The 31-year-old travelled to London from Abu Dhabi, where he had been training with the Russian club during their winter break, and is expected to complete a loan deal until the end of the season, which will help boost an Arsenal squad weakened by Aaron Ramsey's injury setback.
At his scheduled media conference on Thursday morning, Wenger had ruled out a deal for Schalke's Julian Draxler, but confirmed he was looking to bring in at least one other player.
Kallstrom started his career at Djurgarden before joining French club Rennes and then on to Lyon in 2006, where he won the Ligue 1 title twice as well as two Coupe de France.
Having played at the 2006 World Cup and also three European Championships, he left for Spartak in the summer of 2012 and made 10 appearances this season before the winter break.
It has been reported Wenger could also further bolster Arsenal's attacking options with veteran Lazio forward Miroslav Klose, the 35-year-old German international.
However, any suggestion Arsenal were closing in 20-year-old Schalke talent Draxler - valued at some £37million by the Bundesliga side - were rejected by Wenger.
"The Draxler situation has been created by the newspapers, not by me," the Arsenal boss said.
"You ask me, what is that situation? I say nothing is happening on that front. He will stay at Schalke."
Wenger added: "We are open-minded and active. We look at the opportunities.
"We are still trying to bring at least one body in because we are a bit hit now by the red card for (Mathieu) Flamini, the setback of Ramsey and the fact that (Jack) Wilshere is not available, which I hope will be short term.
"We are in an area where we had plenty of players, suddenly we are a bit short. It is 80/20 and I am confident we will be able to do something today."
Arsenal drew at Southampton in midweek and were then knocked off the top of the table when Manchester City romped to a 5-1 win at Tottenham.
The Gunners, though, could regain the lead should they beat Crystal Palace on Sunday, with City hosting third-placed Chelsea on Monday night.
"The results are not made by the winter transfer market, they are made by the performances on the pitch. On that front we have shown we can be very competitive," said Wenger.
"Why should we not (believe)? We are in a position to be able to do it after 23 games with the squad we have available now, so why should we suddenly give up?"
Ramsey, 23, had been one of the stand-out performers in Arsenal's impressive start.
However, the Wales international has suffered a setback in his rehabilitation from a thigh problem.
"It is certainly a few weeks. It will be between four to six weeks. I count six, so I have no bad surprise.
"Of course it's a blow for us because we recently lost (Theo) Walcott for a long period, now Ramsey is out for a long period and it is a big setback."
Arsenal will be without Flamini for three matches following his red card at Southampton, while fellow midfielder Wilshere continues to recover from an ankle problem and should be fit to face Liverpool next weekend.