While Bairstow endured a slow start to his inning, after which Kohli took a swipe at him on Day 2, by taking the name of New Zealand bowler Tim Southee.
The 33-year-old was heard saying on the stump mic: "Little bit faster than Southee, eh?"
Meanwhile, on Day 3, the pair were spotted having a go at each other, with umpires forced to intervene and Kohli signalling Bairstow to shut up.
Talking about the match, England had scored 200/6 by lunch on Day 3, in reply to India's 416, powered by Bairstow's 91-run knock.