The 44-year-old director made the announcement on Twitter, making reference to the anniversary of the music legend's passing in the post, and revealed his father knew about the impending birth before he died.
"1 month since dad died today. Made this card for him at Christmas. Due in June. Circle of life. Love you, grandad," Duncan wrote alongside a drawing of a baby in the womb with an umbilical cord poking out of its stomach.
The BAFTA-award winner is married to photographer Rodene Ronquillo and the baby will be the couple's first and his parents' first grandchild.
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