The Social Network actor, Song, 34, gave birth to their first child, named Dakota, in April 2021. The pair kept news of both of their sons’ arrivals under wraps.
The Independent has contacted representatives of Song and Culkin for comment.
The news comes more than one year after the pair’s engagement was confirmed by US Weekly.
Last year, Song opened up to The Cut about her experiences juggling a new baby, work, and her relationship.
“My boyfriend and I are very hands-on,” she said. “We don’t have a nanny, but my mom has been here with us since my son was born.”
The actor added: “When I was working, my mom would bring him to set so I could breastfeed and see him during the day.”
The pair, who are both former child actors, started dating when they met while shooting the film Changeland. Since then, they’ve maintained a private relationship, except for the occasional Instagram post.
“Happy 40th birthday to this magical being,” Song wrote on Instagram, posing in matching personalised pyjamas with Culkin. “I could sit here and write endlessly about how wonderful, kind, loving, genuine, loyal, honest, brilliant and hilarious you are, and how grateful I am that I get to share and do this life thing with you.”
Song previously starred as Madison Maxwell in the Hulu comedy series Dollface, which was not renewed for a third season.
During filming, Song returned to set to appear in the second season only 12 weeks after giving birth to her first child.
“The biggest challenge is being away from family,” Song said after the experience.
“I know everyone feels that way … I put a lot of pressure on myself to keep pumping and breastfeeding him while I was working. I pumped throughout production.”