The source added: "It's still early, so they are only telling close friends and family members."
Fox is already a stepmother to Green's son Kassius, 10, from his previous relationship with actress Vanessa Marcil.
"Megan has become a great stepmum, and Brian knows she'll be wonderful with the baby," the source added.
"She has really grown in the past couple of years ... her family comes first."
The actress has also spoken of her desire to start a family.
"I've always wanted kids. It's just about the timing being right," Fox told The Hollywood Reporter while plugging her latest flick Friends With Kids in September.
"I have a number in my mind of what amount of money I want in the bank so that I can protect that child's future. And really protect it, so that kid never struggles. Once I feel safe with that then I'll have them."
Check out her video interview with The Hollywood Reporter below:
She also told Access Hollywood at the premiere of the movie earlier this year that she would like "at least two, maybe three" children.
"It's up to God," she continued. "It's gonna happen when it's supposed to happen."
Neither of their reps have confirmed the pregnancy.
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