The Hoops coach has revealed he is preparing to discuss his future with the French striker.
With only a year left on his current contract, it is widely believed that Celtic will be looking to take advantage of Edward, with people like Leicester City keen to sign him.
However, former club PSG are said to be liable for 40% of any potential fees Celtic would receive for the 23-year-old.
And Postecoglou hasn’t ruled out the prospect of persuading Edouard to stay in Glasgow, as he tries to fight for the Premiership title against Rangers.
People like Kristoffer Ajer – the subject of two rejected £ 10million offers from Norwich and Bayer Leverkusen – and Ryan Christie have also been linked with departures from Celtic.
Scottish ace Christie only has six months left on his Parkhead contract.
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But new gaffer Postecoglou has urged any player considering a brutal exit – not to make any rash decisions.
When asked if he refused to rule out the possibility of keeping Edouard and Ajer, he replied: “Why should I when I haven’t had this discussion with them?”
“I will make these decisions myself based on what I see and assess.
“I have a multitude of people giving me advice and opinions. But I am the only one responsible.
“I’m going to have these discussions with the players and make sure the interests of the club are protected.
“Should I make a quick decision on Edouard?” Again, this will be a two-way street. It’s not just about me and what I want.
“Whether it’s Edoaurd or whoever it is, my feeling is that whatever their dreams or aspirations are, I think they can make them come true here at Celtic.
“I hope they see a certain synergy in it and that they want to get involved.
“Because whoever chooses to stay, I want to assure them that they will be a part of something special.
“And maybe they should think about it before they go. Because once you have passed the door, it is very difficult to enter.
“I will do what is best for the club. If that means the players are leaving, they will leave. If that means they stay, they stay.
“But I won’t let anyone use the club as a vehicle for their own purposes.
“When the time is right for this to end, as it will for all of us, let’s hope we leave some kind of legacy.
“But I will not let anyone, player or staff, help themselves and put their personal interests forward.
“So yes, there are discussions that I will be having with the players over the next few weeks.
“But I can assure you that whoever remains will be hired at this club. “
On top of challenges like this off the pitch, Postecoglou is just three weeks away from a massive one, as Celtic face Danish side Midjtyland in the Champions League second qualifying round.
The Hoops team must be rebuilt this summer after the return of several players on loan to their parent clubs.
The future of Edward, Ajer, Christie, Leigh Griffiths, Albian Ajeti and Vasilis Barkas is on hold.
And Euro 2020 players like Callum McGregor, James Forrest and David Turnbull have yet to return to training.
Postecoglou admits he faces a huge task to overtake the Danes and move closer to the Champions League group stage.
But he’s adamant it’s not insurmountable for his new team – and says Celtic fans expect immediate progress.
He said: “Regardless of whether it is too much to expect, people are still waiting.
“No matter what I think, others will measure in their own way.
“I’m sure a lot of people will watch this first game, so that we play fantastic football. Otherwise, they will criticize me.
“What I want to do is make sure we’re ready for the game. It’s going to be a challenge because they are a good team.
“I’ve been following them for several years so I know the team and its structure very well.
“It’s a hell of a challenge for us, but that’s why I’m here. I have three weeks to prepare and each day will be important.
“Is this a challenge too far? No, because every opportunity is a chance for us to show that we can play our football. It’s not the end of the game for me, it’s the starting point.
“Whatever we do in this game, although it’s great, my expectations will be that we have to keep improving. “
Meanwhile, Postecoglou lost another player yesterday as Scotland defender Jack Hendry sealed a permanent transfer to Belgium’s Ostend side, where he was loaned out.
But after winning their Player of the Year award last season, they could be on their way to a quick profit on the 26-year-old, with the West Brom championship team apparently interested.