The No Time To Die star has recalled his conversation with Duchess Catherine at the BAFTAs in February 2019 - when he won Best Actor for his role as Freddie Mercury in Bohemian Rhapsody - and revealed he offered to help as they had just welcomed their third child Prince Louis, now three, in April 2018.
Appearing on Jimmy Kimmel Live this week, he said: "The funniest thing about that was, I said, 'If you ever need a day off, some time off, I'm backup for you'.
"She's like, 'What do you mean?' I go, 'I can babysit, you guys go out, have a good time.'"
The 40-year-old actor felt some sympathy for the duchess - who also has Prince George, eight, and Princess Charlotte, six, with husband Prince William - at the glitzy event.
He added: "What's fascinating is, they work so hard to get to know everybody - who's done whatever film they've done, their past films.
"You can see they've done their homework. And I just looked at Princess Kate at one point, and I said, 'This must be exhausting.' And she said, 'Why?'"
Rami mentioned her baby, and asked how she was doing given the new arrival.
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He said: "[I asked,] 'You just had a baby, right?' I think she was taken aback," the star continued. "She said, 'How are you doing?' [I'm like,] 'No, how are you?'
"And in the most regal, elegant way, she gave me a look. But you can tell. Imagine, you're dressed to the nines, have to talk to all these actors."
And he admitted while the royals have to be "so careful" about what they say, she seemed to be caught "off guard for a second".
He revealed: "They're so careful. But it was so cool. I think I caught her off guard for a second.
"And, you know, had that look of - in the most elegant, professional, royal way - 'Yes, it's a lot, having a kid.' "