The Board of Control for Cricket in India (BCCI) invited applications for the role last month and was also considering renewing Kumble's one-year contract.
The 46-year-old former Indian captain, who was put in charge last June, had an impressive stint as a coach, leading India to 12 victories from 17 tests and losing just once.
He also led India to one-day international series wins over New Zealand and England with losses coming only in a Twenty20 series against the West Indies and at last week's Champions Trophy, where they lost to Pakistan.
The Indian team are set to play a five-match ODI series and a T20 game in the Caribbean starting on June 23.
No one at the Board was immediately available to comment.