The 33-year-old actor, currently promoting his new film 'The Danish Girl' is said to be "extremely excited" about starting a family.
The couple have not yet confirmed the news.
A source told The Sun: "Eddie and Hannah are delighted and extremely excited.
"They don’t want to find out the sex of the baby but are happy to tell close friends and family they’re expecting."
The source added it had been a "year to remember" for Redmayne - winning an Oscar and now becoming a dad. The baby is reportedly due next June.
Redmayne married Bagshawe in December 2014 in a small ceremony after dating for two years.
A representative for Redmayne has been contacted for comment.
Congratulations to the couple!
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