Finch suffered cartilage damage in St Lucia last month and missed the tour of Bangladesh.
The opening batsman went under knife on Friday and the expectation is that he will be ready to lead his country when the T20 World Cup – staged in UAE and Oman – gets under way in October.
A Cricket Australia statement said: “His recovery is expected to take 8-10 weeks meaning he should be available for the opening games of the World Cup in mid-October.”
Australia will face England, South Africa, West Indies and two yet-to-be-determined qualifiers in Group 1 of the Super 12 stage of the T20 World Cup.
Matthew Wade stepped in to skipper an under-strength Australia side in a 4-1 T20 series loss to Bangladesh.
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