Giakoumakis remained at Celtic at the end of the transfer window but different deadlines in Japan and the United States mean he can still wrap up a move to suitors such as Urawa Red Diamonds and Atlanta United.
"It will get concluded fairly soon," Postecoglou said at a media conference.
"To be honest, it's not on my radar any more. I will probably hear it about the same time as you guys. Maybe, if you are on Instagram, you will hear about it earlier than me.
"It's not on my radar, I don't expect him back in the group, but it hasn't been finalised yet."
The Greece international arrived at the club in the summer of 2021 from Dutch side VVV-Venlo.
The 28-year-old went on to make 57 appearances for Celtic across all competitions. He also scored 26 goals during his spell in Glasgow.