"They've been laying low over the festive period so they can get used to having a busier household and adjusting to the needs of a newborn.
"They love the name Forrest as it's earthy and strong. Tom has always found the movie 'Forrest Gump' very inspiring and enjoyable and thinks it fits."
As of the time of writing, neither Tom nor Charlotte have commented on the reports.
Meanwhile, the Venom star admitted in August last year that he doesn't think there's a "harder job on the planet" than parenting.
He said: "There is no harder job on the planet, and more important, than parenting.
"You've got the military, police, doctors, service personnel - massive respect, huge consequences - but parenting? It's beyond a job, isn't it."
Prior to the birth of their second child together, Tom and Charlotte, 37, relocated from their mansion in south west London to a more rural area on the outskirts of the city after being pestered by an obsessive female fan, as they wanted privacy for their new arrival.
A source said at the time: "Tom is living his new life in the country and has been seen dog-walking every morning. He feels it's the perfect place for him and Charlotte to bring up their new baby."