The 29-year-old hurt his left leg running between wickets during a match against India on Friday, eventually limping off the field after being run out.
"Aaron had scans on his injured left hamstring that confirmed a moderate grade hamstring injury," sports medicine manager Alex Kountouris said.
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"Although this current injury doesn't require surgery there is a period of rehabilitation required before he can resume playing."
Finch is set to miss the one-day series in New Zealand which begins on Wednesday, but Kountouris said it was not a recurrence of an old hamstring injury which needed surgery in 2015.
"We are aiming to have Aaron fit for the ICC World T20 in India but will have a better idea about his fitness for that tournament in the coming weeks once we see how he responds to treatment and a rehabilitation programme," he said.
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The top-order batsman had little doubt, however, tweeting: "I'll be in."
India will host the World Twenty20 from March 8 to April 3, with Mumbai and New Delhi staging the semi-finals and Kolkata the final.
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