Mr Jones announced the news in a tweet, which included a hand-drawn cartoon of a baby inside the womb along with a message referencing his father's death.
"One month since dad died today. Made this card for him at Christmas. Due in June. Circle of life. Love you, granddad," the director, 44, wrote.
Mr Jones is the eldest child of Bowie, who died of cancer last month, aged 69.
The Bafta award-winning director is the singer's only child from his 10-year marriage to his first wife, Angie.
He is married to photographer Rodene Ronquillo.
David Bowie's son Duncan Jones has announced he is to become a father for the first time.