The Board of Control for Cricket in India (BCCI) said on Sunday that it had received 57 applications for the post but did not reveal any names.
Former team director Ravi Shastri, current chief selector Sandeep Patil and ex-pace bowler Venkatesh Prasad were among a few top former players who made their coaching intentions public.
But on Monday Kumble, who has mentored Indian Premier League sides such as Mumbai Indians and Royal Challengers Bangalore, was reported to have also thrown his hat in the ring.
"Yes, I have applied," the former India skipper was quoted as saying by The Indian Express newspaper.
The 45-year-old bagged 619 test wickets in 132 matches and is only the second bowler in cricket history after England's Jim Laker to have taken all 10 in a test innings.