The royal interviewed the primatologist for the latest edition of British Vogue, which his wife the Duchess of Sussex is guest-editing.
Amongst discussing the environment and the human condition, he admitted he views the natural world differently since becoming a father to three-month-old son Archie.
When Dr Goodall said there would be conflicts “over the last fertile land, the last fresh water”, Harry issued a dire warning about the consequences of mankind’s actions.
He said: “What we need to remind everybody is: these are things that are happening now. We are already living in it. We are the frog in the water and it’s already been brought to the boil. Which is terrifying.”
He added: “I think, weirdly, because of the people that I’ve met and the places that I’ve been fortunate enough to go to, I’ve always had a connection and a love for nature.
“I view it differently now, without question. But I’ve always wanted to try and ensure that, even before having a child and hoping to have children.”
When Dr Goodall added, laughing: “Not too many,” the duke replied: “Two, maximum.”
During the discussion, Ms Goodall said: “It’s crazy to think we can have unlimited economic development on a planet with finite natural resources.
“There’s a lot of violence and war and suffering around the world today, but we’re part of the natural world, and if we can’t learn to live in harmony with it, then this is going to get worse.”
When Harry asked how her study of primates had impacted upon how she felt about people, the discussion turned to the issue of violence and racism.
Dr Goodall said it was “obvious” mankind had “inherited aggressive tendencies”,but human brains were able to control anger.
The duke said the same applied to“unconscious bias”, where someone’s words or actions could be perceived as racist,but if confronted the person would deny it.
He added: “I’m not saying that you’re a racist, I’m just saying that your unconscious bias is proving that,because of the way that you’ve been brought up, the environment you’ve been brought up in, suggests that you have this point of view…”
Dr Goodall’s lifelong work with chimpanzees has made her an international figure, and she is credited with making the first recorded observations of chimpanzees using tools and eating meat.
The conservationist welcomed Harry to a meeting of her youth empowerment project Roots and Shoots earlier this month at Windsor Castle.
She was also invited to the duke’s Frogmore Cottage home near Windsor Castle in June where she cuddled Archie, who was then five weeks old.
As a childhood hero of both the Duke and Duchess, Ms Goodall was asked to take part in the special edition of Vogue which features 15 female trailblazer, including New Zealand Prime Minister Jacinda Ardern and student climate change activist Greta Thunberg.
– The September issue of British Vogue is available on Friday August 2.