The midfielder is set to end his eight-year association with French football and continue his career in Russia, subject to putting pen to paper and completion of a medical
The Sweden international will now undergo a medical in the Russian capital before signing a contract, although the details of which remain unspecified.
Jean-Michel Aulas, Lyon's president, also revealed that the deal with Spartak had been sealed. He added that the amount of the transfer is €3 million, "but [the figure] is not important. The most important thing was to remove a big salary," he explained prior to the club's clash with Montpellier on Saturday.
Kallstrom, 29 arrived in Ligue 1 with Rennes via Djurgarden in 2004, before moving onto then-champions Lyon just two years later.
He made 171 appearances for Les Gones, scoring 17 times, but his time at club appeared to be up following president Jean-Michel Aulas' admission that he was ready to move the midfielder on.
The Swede joins the likes of Unai Emery, Romulo and Juan Insaurralde as new faces at Spartak, who have spent the summer rebuilding their squad in an bid to end Zenit St Petersburg's hold on the Russian Premier League crown.