The 28-year-old has had his injury problems this season but when fit, he has struggled to force his way into previous boss Alan Pardew's plans - making three substitute appearance in the Premier League, the last of which came against Southampton at the start of December.
Whether new manager Sam Allardyce has bigger plans for the Mali international remains to be seen, the former England boss naming him on the bench for Sunday's loss at Arsenal.
Birmingham are now set to test the Eagles' resolve to keep the former Wolves man, reportedly lodging a significant offer for Sako as they seek to add to their attacking options.
Blues are not the only club interested in Sako's services though, as their Championship rivals Derby are also believed to be monitoring the situation.
One stumbling block to any deal going ahead could be Sako's participation in the upcoming Africa Cup of Nations, with Mali's campaign getting underway on January 17.