Southee felt pain in his front foot while bowling in training, with a scan showing a grade two ligament strain in his left ankle, according to the New Zealand Cricket website.
He will return home for recovery but is expected to be fit for the subsequent five-match ODI series against India. Southee's injury is a setback for New Zealand as the 27-year-old is their most experienced Test bowler, having taken 177 wickets in 52 Tests at an average of 32.63. Matt Henry, who has played four Tests, will replace Southee in the squad.
"The focus now is for Tim to give his ankle 7-10 days rest, before slowly building his bowling loads back up ahead of the ODI series.
"We've got a replacement who is ready to go in Matt Henry and he'll join the team before the start of the first Test," he added.
New Zealand are currently playing their only warm-up match of the tour at the Firoz Shah Kotla.
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