Goal understands that Arsenal has agreed a deal to take Kallstrom until the end of the season and pay a loan fee to Spartak.
The Londoners are hopeful of signing the Swede before the transfer deadline as they close in on two new signings.
The club is also negotiating a deal with Lazio to take striker Miroslav Klose on loan until the end of the season. Arsenal contract fixer Dicky Law is in Italy to finalize a deal for veteran German striker Klose.
The Gunners have lined up Real Madrid youngster Alvaro Morata as an alternative if they cannot finalize terms with Lazio for Klose but they are increasingly confident of landing the prolific 35-year-old.
With Aaron Ramsey out for up to six weeks after sustaining a fresh thigh injury in training, Wenger decided to reinforce his central midfield.
The Arsenal boss moved for Kallstrom after learning that first-choice target Blaise Matuidi will extend his Paris Saint-Germain contract and being put off by Southampton’s valuation of Morgan Schneiderlin.