The Greece international has managed only one goal in eight appearances since joining from Dutch side VVV-Venlo in a deal worth up to £2.5m on transfer deadline day.
He was one of 10 Celtic players who missed the 0-0 draw with St Mirren on Wednesday with a dearth of attacking options resulting in winger Liel Abada starting at centre-forward.
Giakoumakis been out of action since November 7 after picking up a knee injury and playing through the pain barrier against Dundee. The 27-year-old required a minor operation to clean up the knee and is now nearing a return.
Postecoglou confirmed that Giakoumakis will return to training this week but that he will not be ready for the Boxing Day clash with St Johnstone in Perth.
“Jacko is really close. He will probably train with the team in the next session,” the Celtic boss said.
“However, one session will probably not be enough for him to be right for Sunday. He is back now to almost being available.
“If we had one more game he would probably be available but we are now looking to get him back after the break.
“It’s unlikely he’ll make the bench, that would be a stretch. He is finishing off his rehab today and then we have one session on Christmas Day so unlikely.”