The 23-year-old signed from Paris St Germain in 2018 for a reported fee of £9millon, after an initial loan spell at the Glasgow club, and his contract is up at the end of the season.
Despite speculation linking the French striker with Brighton, Hoops boss Postecoglou is not aware of any concrete bids.
Reports claim Edouard is prepared to let his contract expire, leaving him free to speak to other clubs in January, and the former Australia boss admits the prospect of losing such a valuable asset for nothing will be a worry for the club.
Ahead of the second leg of their Europa League qualifier against FK Jablonec on Thursday night, which the Hoops lead 4-2 from the first leg in the Czech Republic, Postecoglou said: "I think you are in right in saying that is a club concern rather than my concern right now.
"No, (I'm) not concerned at all.
"If it means he stays here and contributes then so be it, but again that is stuff that I am not really spending too much time thinking about.
"I have been at pains to say that there are certain things that are out of my control that I'm just not going to waste time during this period anyway, where it is really hectic, thinking about it.
"My total focus is getting the team ready for these games, playing our football which is not easy and needs a lot of attention from me and the staff, and also working on bringing players in.
"I have left outgoings totally to other people, that's other people's concerns.
"You are probably right, there is a concern there around cashing him in, but those are probably questions for other people."
Postecoglou's squad has a clean bill of health, including Edouard, who picked up a knock in a post-match training session following the 6-0 cinch Premiership win over Dundee at the weekend.
He said: "I reckon that falls into the remit of not in my world right now.
"I have met Gordon and he is a very passionate football person which I kind of like.
"The information I have had is that we are keen to tap into his knowledge of the club in terms of some of the structures around the club, particularly around the academy and women's football which I think is really important infrastructure-wise.
"But again, how much that comes on my radar at the moment, this is probably the first time I have thought about it."
Celtic Park will be full for the first time since before the pandemic for Thursday's match and Postecoglou wants his side to come out "all guns blazing".
He said: "When the whistle goes I want to show the same intent we showed at the weekend.
"Irrespective of what the opponents may or may not come up with, it is important we set the tempo right from the start and let them adjust to us."