The wing-back's season is over after suffering a setback this week.
Postecoglou said: "A disappointing one for Destiny and for ourselves as well. He's been a big part of our year. It was an unfortunate incident in training. He's had the operation, it's gone well so hopefully that gives him time come pre-season to get back into it. We will miss him.
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On other team news ahead of facing Arsenal, he added: "Skippy (Oliver Skipp) picked up a small knock in training so he will miss this weekend.
"They are the outs, apart from the long-term ones. Pedro (Porro) and Richy (Richarlison) both trained all week, so they are both available."