The BBC Radio 5 Live host took to Twitter to reveal that he and partner Rebecca McFarlane will welcome the child later this year.
Bacon told his followers: "We're having a baby. Our first one. What's this whole fatherhood business like then? Are babies allowed in The Groucho?"
He also confirmed that his wife is 14 weeks pregnant, although he did not reveal the sex of the baby.
The former Blue Peter presenter married McFarlane in 2008 after meeting at Capital Radio and dating for 18 months.
In 2008, he underwent a throat operation to remove a growth from his vocal chords.
Last year, he announced his resignation from BBC Radio 6 Music, in order to concentrate on "increased work commitments".