This happened when the couple chatted in a live Instagram session after midnight in Kyrgios' native Australia - which was the middle of Saturday afternoon for Murray in the UK.
"Honestly, I think you are better than Djokovic in my opinion," said Kyrgios. "Djokovic plays dodge on my serve. He couldn't fire him. And you're there ... slapping him for a winner. I couldn't get him past you."
Kyrgios is 2-0 against Djokovic while losing five of the six to the Scot. But Djokovic won 17 Grand Slam titles against the three of Murray to make him one of the three great men in tennis alongside Roger Federer and Rafael Nadal.
Although Kyrgios and Murray have always gotten along - even making a roller coaster together in Cincinnati in 2018 - the world number 40 has not hidden its disdain for Djokovic.
He declared last year during a podcast that the world number one had "a sickly obsession with wanting to be loved" while dissidenting Nadal.
Kyrgios has already been fined and suspended for some of his antics in the past, but has helped jumpstart tennis relief efforts for the bushfires that ravaged Australia before the January Australian Open where he lost to Nadal.
The interruption of tennis due to the ongoing coronavirus pandemic has seen Murray - who celebrated his 33rd birthday on Friday - participating in Instagram lives with Nadal and Djokovic.
Nadal, however, denied a request from Kyrgios, 25, to chat on the social media platform.
Murray will be back
His career saved by hip resurfacing last year, Murray suspected he would be ready to return to action when the professional tennis circuit resumed.
"I think when things get back to normal, I think I'll be fine," said Murray.
Kyrgios wants to play doubles with him.
But a return date is not yet known. And after the elimination of clay tournaments leading to the French Open and then to European turf competitions including Wimbledon, the ATP announced Friday a new suspension of the men's tour until the end of July.
The cancellation of Wimbledon marked the first grand slam of this year to be officially canceled. Next month, the fate of the US Open at the end of August should be determined.
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The French Open, meanwhile, is now scheduled for September after having controversially moved its place on the calendar.
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