As India prepare to launch their tournament against Pakistan in Birmingham on Sunday, Indian media is abuzz with reports of differences between Kohli and Kumble over tactics and training.
With Kumble's one-year contract ending this month, the Indian board's move to advertise the job ahead of the 50-over showpiece event raised eyebrows.
The deadline for applications ended on Wednesday. However, batting legend Sunil Gavaskar believes the Board of Control for Cricket in India (BCCI) was just "following procedure".
Gavaskar, attempting to reassure fans, said an invitation for applications did not reflect tensions in the Kohli-Kumble relationship.
"You will never have a situation when the captain and coach of any country is on the same page," Gavaskar said at the Aaj Tak Salaam cricket convention in London on Wednesday.
"It is impossible for that to happen because the coach had played in the earlier generation and that's why the approach may be slightly different from the current generation."
"It might not be something actually happening on the field – it could be something to do with practice sessions or team combinations," the former captain added.
"I don't think that should be taken too seriously because these discussions are healthy for the team."
Kumble 'has done nothing wrong'
Indian newspapers have speculated that cracks in the Kohli-Kumble relationship prompted the BCCI to hunt for a new coach.
The Times of India reported that players were not happy with Kumble's hard task-master approach, and that the captain and coach had several run-ins during this year's home series against Australia.
A headline in The Hindu read "Kohli, Kumble at loggerheads?" with the report adding that Kumble's proposal to field left-arm wrist spinner Kuldeep Yadav in the third Australia Test in Ranchi set off trouble between the two.
Yadav was finally included in the playing XI in the fourth and final Test, with Ajinkya Rahane leading the side in place of an injured Kohli.
Despite the apparent tensions, Kumble has had a brilliant run as coach, helping the Kohli-led side win all five Test series since his appointment in June last year.
"Kumble has done a very good job as coach. I am simply going by the results and when you go by the results of last year, you can say he has done nothing wrong," said Gavaskar, the first batsman to breach the 10,000-run mark in Tests.
It was also reported that the advertisement of the India coaching job has met with a lukewarm response, with few big names coming forward.
While Kumble is an automatic candidate for a new contract, the Times of India said Australia's Tom Moody and New Zealand's Daniel Vettori were also being discussed.