Kumble, Amre and Rajput met the panel, comprising Sourav Ganguly, VVS Laxman and Sachin Tendulkar, the latter joining in via video conference from London.
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Former BCCI secretary Sanjay Jagdale, who is 'coordinator' of the selection process, assisted the trio.
It was learnt that former India team director Ravi Shastri, the other high-profile candidate for the job, was also interviewed via video conference as he is away.
The BCCI received 57 applications from both India and overseas, from which the list was trimmed to 21.
The committee is expected to forward its recommendations to the BCCI president by June 22 and finalise the candidate by June 24.
IMAGE: Mumbai Indians mentor Anil Kumble during IPL 2015
Photograph: BCCI