McKenzie, whose contract with New Zealand Rugby is due to finish at the end of the year, played 95 times for the Chiefs and is reportedly set for a lucrative move to Japan.
"All I know is Damian is not going to be with us, at the moment that's all I can sort of comment on," the Chiefs chief executive Michael Collins told Stuff.
"I think Damian McKenzie's probably irreplaceable. He's one of the Chiefs' favourite sons, done a great job for us over many years."
The Chiefs on Tuesday welcomed the arrival of Māori All Blacks first-five Josh Ioane from the Highlanders.