Root's backing comes in the wake of Cook's recent comments that he could quit at the end of the ongoing five-match Test series as he was finding it hard to be apart from his family for long stretches.
"I think he is pretty set on captaining for a few years to come and I hope he does because I think he is a brilliant leader and I've really enjoyed playing under him," Sport24 quoted Root, as saying.
The Test vice-captain, who is widely tipped to succeed Cook, also dismissed talks that he was ready to take the helm and insisted that he was happy with his role in the team.
"I'm quite happy with the role I'm playing at the moment and you want to do what is right for England and the side," he said.
Cook, who recently overtook Michael Atherton's record of 54 Tests as England captain, no longer plays in limited-overs format but has been a permanent member in the Test side.
While England are trailing by 0-2 in the five-match series in India, Cook's own form has held up as he scored his 30th Test century in the opening match in Rajkot.
"And genuinely, honestly, I think he is the right man. Hopefully, we get these results we want in the last two games and we can put right what has been quite a frustrating last couple of weeks."
Root, who himself averaging nearly 50 in India, remained hopeful that Cook would cope well in the remaining Tests, while also calling on the other batsmen to step up their game.
England must have to win their next match beginning on December 8 in Mumbai, to hold any chance of saving the series.
(This story has not been edited by Business Standard staff and is auto-generated from a syndicated feed.)