The former Australia head coach has been linked with the vacancy in Yorkshire as the Premier League club seek a replacement for Jesse Marsch.
Postecoglou sealed the cinch Premiership title and League Cup in his first season in Scotland and his attack-minded side are nine points clear of Rangers in the league as well as pushing for success in both cups.
Former Middlesbrough and Fulham goalkeeper Schwarzer told Sky Sports: “I think he would be an exceptional fit. But do I think he is going to leave Celtic and go to Leeds right now at this moment of time? I don’t think so.
“Not that I have spoken to him, I am completely speculating here, but I don’t think personally he would leave Celtic, if he were going to leave at all, at this particular moment in time. His relationship, his rapport with the club, with the fans is too important to him.
“Also I think the job he is doing at Celtic is phenomenal, if anything it’s a little bit underrated how good a job he is doing, considering where Celtic were and how far they were behind Rangers only two seasons ago and where they are right now, clearly the best side in Scotland.
“If Ange is ever going to leave the club it will be during an off-season, it will be done amicably, it will be done in the right manner, because that’s how he does business.”
Rayo Vallecano head coach Andoni Iraola and Feyenoord’s Arne Slot emerged as favourites for the Leeds job after another reported target, Carlos Corberan, signed a new contract with West Brom on Tuesday night.