Kumble has guided the team to just one defeat in 17 tests since taking charge, and the BCCI has made it clear that it will also consider extending his contract.
"There are no issues whatsoever," Kohli told a news conference on Saturday. "There has been a lot of speculation and a lot of things written by people without actually being a part of the changing room, which is very strange.
"There are no problems whatsoever. The team is totally focussed on the Champions Trophy."
Kohli said differences of opinion were part and parcel of the game, and that the board's invitation for applications was a routine matter.
"In any walk of life there are agreements and disagreements always," he added. "These are normal things. Even at home you do not agree on everything with your family.
"If something is put in place as a process, I don't see why people are creating so many speculations about it. It has been followed last time as well, and I didn't see any issues being created last time. So it is the same process.
"It is just happening after 12 months."
Pressed on his own relationship with Kumble, Kohli said he was enjoying working under the former India captain.
"It's been really good. The whole journey has been good," he added.
(Reporting by Simon Jennings, editing by Pritha Sarkar)
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