Portuguese winger Jota has already been ruled out with a hamstring injury and the Hoops boss had previously said Swiss striker Albian Ajeti, also suffering with a hamstring problem, would probably not be back before the winter break while Greek forward Giorgos Giakoumakis had a clean-up operation of his knee.
Japan international Kyogo Furuhashi (hamstring), James Forrest and Mikey Johnston (both knocks) also missed the midweek cinch Premiership win at Ross County and Postecoglou will give them every chance to make the showpiece occasion.
He said: "From the other night, everyone got through OK which is good.
"In terms of guys in rehab, all are progressing.
"Obviously we still have another day tomorrow but can't really make a call on any of them being available or unavailable at this stage.
"As I said earlier in the week, it is a day-to-day proposition as to who potentially might be available for Sunday."
Postecoglou also revealed that defender Christopher Jullien is training but is still not ready to play.
The Frenchman has not featured since rupturing his cruciate ligament against Dundee United in December last year.