Speaking on her new Archetypes podcast, Meghan spoke of her frustration of having to attend a royal engagement after her son Archie's nursery caught on fire during a visit to South Africa.
Her son was not in there at the time, but Meghan said she was "shaken".
She also spoke about the challenges of being a woman with ambition.
"I don't ever remember personally feeling the negative connotation behind the word ambitious until I started dating my now husband. Apparently, ambition is a terrible, terrible thing, for a woman, that is," she said.
She added on the podcast some believed ambitious women "have an agenda".
"You must be calculating or selfish or aggressive, or a climber," she said.
Speaking to Sport and Aged Care Minister Anika Wells about the podcast, Ally admitted Meghan rubbed her the wrong way and there was "just something" she didn't like.
"II really try to like Meghan Markle; I think some of the arguments she makes about ambition are very true, but there's just something, you know?" Ally said.
Elsewhere on Today, host Karl Stefanovic also had a dig at the Duchess, sneering at her reaction to her son's fire scare.
"So, just to get this right, the baby's, what, not even there?" he said.
When Ally explained the nanny had taken Archie from the bedroom prior to the fire, Karl struggled to understand why the Duchess was so rattled.
"The baby's not there when the fire catches, right? And because the baby wasn't there, she was going to have difficulties going to a function? Oh, my God," he said as he rolled his eyes.
Ally tried to defend Meghan and said she was a first-time mother in a foreign country.
"That is not you talking. You're a woman who says, 'Grow a set', that's what you would say normally," Karl hit back as Ally nodded in response.
Meghan Markle's new podcast has been widely criticised, with The Times columnist Hilary Rose dubbing it "pure, narcissistic gibberish", with other columnists in the UK just as scathing.
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