McGree is a player well known to Postecoglou from his time with the Australian national team, but he is expected to wrap up a move to Teesside - and not Glasgow - shortly.
The Celtic manager said: "I can tell you there are no other offers out there, we haven't spoken to anyone else. Riley McGree was somebody we had an interest in because of my association with him, obviously I know the player very well.
"After initial discussions, he's decided to look elsewhere, which is fine."
Meanwhile, Boro's glamour FA Cup fourth round tie at Manchester United has been selected for live TV coverage.
The game will be played on Friday February 4 (8pm) and will be screened on ITV.