Defeat in the Europa League to FC Midtjylland left Postecoglou winless in his first six games, which makes him the first permanent manager in 100 years to experience such results.
Speaking after the loss, Antonio said on BBC Radio: "I feel he will get extra time because he was appointed by the owner.
"He will give him extra time to get his philosophy going, because right now it feels he is struggling to get his tactics across to the players."
Former Forest and Leicester boss Martin O'Neill added: "Wins are the name of the game - it is as simple as that.
"The crowd were already against him in many aspects, but we are six games in, you'd hardly know the name of the tea lady after that amount of time."
On the boos, O'Neill added: "You have to deal with all the noise around you. You can't allow it to dictate everything, but you can't avoid it."