Lehmann, 54, played 27 Tests for Australia. He was Australia's coach for four of the Ashes series in which Root played.
When Lehmann was playing in Yorkshire, where he played overseas until 2006, Root was just coming into his own.
Lehmann will move to the district league team Northamptonshire as head coach for the 2025 season.
“I haven’t spoken to him in a while,” Root added.
“I will do my job and he can do his job. He has complete right to say whatever he wants. That’s his right.”
Lehmann later wrote down his initial comments
“He will be a great one, I know that, and a wonderful ambassador for our game, but he needs that and he will probably do that in the next Ashes. Get a hundred in Australia and then there will be no dispute.”
Root overtook Sir Alastair Cook in the first Test in Pakistan in October to become England's all-time leading run-scorer. With a tally of 12,754, he is fifth on the all-time list, behind Sachin Tendulkar, Ricky Ponting, Jacques Kallis and Rahul Dravid.
Root, who regularly features in the 'Fab Four' of Test batting alongside Kohli, New Zealand's Kane Williamson and Australia's Steve Smith, has been enjoying the best form of his career since the start of 2022.
During this time, his 3,223 runs are significantly more than any other player in Test cricket and include 12 hundreds.
In Christchurch he will become only the fourth Englishman to play 150 Test matches, following his former team-mates Cook, James Anderson and Stuart Broad.
“It's a lot of field work, isn't it?” he said. “It’s really nice to have played so many games and still be involved.”
“In many ways it feels like just another game. I have a lot more to offer this team and this jersey.”
“It will be nice to try and start the series well and get us off to a good start.”
The first game of the three-Test series begins on Thursday (22:00 GMT, Wednesday) at Hagley Oval.
