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.

The hosts, who were knocked out of the FA Cup by League One side Rochdale on Saturday, fell behind in the 14th minute when winger Ben McKenna made a penetrating run down the right flank and produced a neat finish.
Barrow, having been watched by a crowd of 4,414 at Holker Street for the cup tie, were looking to improve on sixth place in the National League table, and went close to an equaliser before half-time when an acrobatic effort by defender Moussa Diarra hit the crossbar.
Southport could have extended their lead in the second half, but Andrai Jones' close-range effort was saved by Jonathan Flatt as the visitors held out to secure a crucial three points in their battle to pull clear of the relegation fight.
Report supplied by the Press Association.
Match ends, Barrow 0, Southport 1.
Second Half ends, Barrow 0, Southport 1.
Substitution, Southport. Jake Wright replaces Delial Brewster.
Substitution, Barrow. Lindon Meikle replaces Jordan Williams.
Substitution, Barrow. Ross Hannah replaces Richard Bennett.
Substitution, Barrow. Andy Haworth replaces Dan Rowe.
Second Half begins Barrow 0, Southport 1.
Jordan Williams (Barrow) is shown the yellow card for a bad foul.
First Half ends, Barrow 0, Southport 1.
Goal!  Barrow 0, Southport 1. Ben McKenna (Southport).
First Half begins.
Lineups are announced and players are warming up.
David Bowie's son Duncan Jones has announced he is to become a father for the first time.
[NEXT_CONCEPT]
Barrow suffered a narrow home defeat by Southport on their return to National League action.