"As it seems now, I'm likely to take over as the team captain from Pilic, who will stay on as my adviser," the 34-year-old told a local newspaper.
World No 15 Marin Cilic is under the radar of the International Tennis Federation for failing a drug test during the BMW Open in Munich in April, it has been reported.
Kyrgios shot back at Coric, saying he "should care" about his comments.
Croatia achieved their best-ever result in the tournament in losing Sunday's match to France, surpassing the third place finish of 1998.