Shastri, 55, was among five high-profile candidates interviewed on Monday for the post left vacant by ex-Test captain Anil Kumble, who resigned amid difficulties over his relationship with Kohli which he described as "untenable".
An advisory committee of Sachin Tendulkar, Sourav Ganguly and VVS Laxman has been tasked with selecting the new coach to head the world's number-one ranked Test team.
The cricketing greats considered presentations from Shastri, Virender Sehwag, Australia's Tom Moody, Englishman Richard Pybus and former India manager Lalchand Rajput.
But after a gruelling round of interviews the committee decided there was "no hurry" to name the coach, and felt Kohli should be consulted before the final decision was made.
Speculation has been rife that Shastri, who served as India's team director between 2014 and 2016, is on friendly terms with Kohli and will therefore get the nod.
"All members of the Indian cricket team are in favour of cricketer-turned-commentator Shastri taking over the role," reported The Times of India on Tuesday.
"All 15 members of the team are unanimous on this. It's not even 14-1," the daily reported, citing an unnamed member of the Indian cricket team.
Ganguly told reporters that Kohli would be briefed on the developments once the team returned from their limited-overs tour of the West Indies.
"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," said Ganguly.
"We want to speak to Virat Kohli once he is back from America, all three of us along with the respective people concerned.
"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."
Kohli made no recommendation for who should fill the coaching role, Ganguly added, which would run until the World Cup in 2019.
Shastri, who has represented India in 80 Tests and 150 one-day internationals between 1981 and 1992, originally applied when a one-year contract to coach India was made available in 2016.
A decision by India's cricket board to allow Shastri to interview via Skype while on holiday at the time angered Ganguly, who complained publicly about the selection process. Shastri was then pipped to the post by Kumble.
