Midfielder Kallstrom, 31, is set to miss at least six games.
The Swede was hurt on Tuesday while playing football on a beach during Spartak’s pre-season tour in Abu Dhabi.
He told Arsenal of his discomfort on arrival at the club on Friday but despite an MRI scan showing no damage, a CT scan found an issue with his back.
The Arsenal medical team provided manager Arsene Wenger with the prognosis, but it was decided Kallstrom would still join and undergo his rehabilitation with the Gunners.
He is expected to miss at least Arsenal’s next six matches, in which they face Crystal Palace, Liverpool twice, Manchester United, Bayern Munich and Sunderland.
Kallstrom did not train with his new team-mates on Friday or Saturday and will briefly return to Sweden and Russia before moving to London and beginning a course of treatment.
The former Lyon player, Arsenal’s only signing in the January transfer window, joined on loan until the end of the season to provide cover for injured midfielders Aaron Ramsey and Jack Wilshere, plus the suspended Mathieu Flamini.