Gunners manager Arsene Wenger has made the approach after admitting defeat in the race to sign Schalke’s Julian Draxler, and hearing the news the Aaron Ramsey will be ruled out for up to six weeks.
Kallstrom moved to Russia in the summer of 2012 having spent six years in France with Lyon, where he won both the Ligue One title and Coupe de France on two occasions.
Internationally, the 31 year old has made 108 appearances for Sweden, scoring sixteen goals.
Arsenal, who were knocked of the top of the Premier League by Manchester City midweek, have also been linked with a move for veteran Germany striker Miroslav Klose.
The North London club can return to the summit with victory over Crystal Palace at the Emirates Stadium on Sunday afternoon, and they can be backed to win without conceding at 8/11.