Former captain Kumble stepped down as coach last month after falling out with skipper Kohli despite enjoying a successful run during his year in charge.
Last year the BCCI formed the CAC, comprising former players Sachin Tendulkar, Sourav Ganguly and VVS Laxman, who chose Kumble for the job.
The CAC interviewed Indians Ravi Shastri, Virender Sehwag and Lalchand Rajput, Australian Tom Moody and Englishman Richard Pybus to fill the vacant position, the BCCI said in a statement.
India toured West Indies for a limited-overs series without a head coach and could continue with the set-up for their visit to Sri Lanka, starting on July 26.
"We feel there is no hurry at the moment," former India captain Ganguly told reporters.
"Sri Lanka tour is in a week's time, the board headed by (secretary) Amitabh Choudhary and (Chief Executive) Rahul Johri will probably continue with the same set for the time being."
Kumble's original contract expired with the India's Champions Trophy campaign, but the former leg-spinner was tipped to get an extension after leading the country to 12 victories in 17 tests with just one loss.
Kumble, India's most successful bowler, also guided the team to one-day international series wins over New Zealand and England with losses coming only in a Twenty20 series against the West Indies and in the Champions Trophy final, where they lost to Pakistan.
"We want to speak to Virat Kohli once he is back from America, all three of us along with the respective people concerned," Ganguly said, adding that Kohli has made no recommendation.
"We will explain to him that the coaches want to function in a certain way and make sure that everybody is on the same page before we make the announcement, because once we make the announcement it has to be till the (2019) World Cup."
Former test all-rounder Shastri was appointed team director after head coach Duncan Fletcher's term came to an end following India's semi-final defeat at the 2015 World Cup in Australia and New Zealand.
Moody has previously coached the Sri Lankan national team and in the Indian Premier League while Pybus had roles with the Pakistan, Bangladesh and West Indies teams.
(Reporting by Sudipto Ganguly; editing by Ed Osmond)
