The paceman has been troubled by the ankle for a while but decided to have an operation after he was forced out of the forthcoming tour of India with a strained stomach muscle.
Team physiotherapist Errol Alcott said: "With the injury to his stomach muscle, we were presented with a window of opportunity to also correct this problem while Brett takes a break from competition.
"Brett will be reviewed again towards the end of November following an intensive rehabilitation programme."
World champions Australia and New Zealand take part in a triangular one-day series in India which starts on Thursday.
Lee felt pain in the back of his ankle during this month's two-test series against Zimbabwe but Alcott said the injury should heal quickly after surgery.
Australia's bowling stocks have been depleted by injury in the past month with pacemen Glenn McGrath and Jason Gillespie and leg spinner Stuart MacGill all pulling out of the tour of India.