Bradley Cooper’s bachelor pad may be plenty big for him, but is it fit for a nursery?

The actor, 41, and his girlfriend, Irina Shayk, 30, are expecting their first child, and the Los Angeles property where the two are reportedly living isn’t exactly baby-proof, a source tells PEOPLE.

“It seems L.A. is now their home base,” the source says. “When they are together in LA, they are very low-key and mostly hang out at home.”

Cooper scooped up the Pacific Palisades home in December 2009 for $4.7 million. Although the 3,335-square-foot estate might seem like enough room for his and Shayk’s new arrival, the single-guy digs would reportedly need extensive renovations to be safe enough for a child.

WATCH THIS: Inside Bradley Cooper & Irina Shayk’s Surprise Baby News

The dad-to-be definitely has baby on the brain, though, with a source telling PEOPLE he’s been “more protective,” of Shayk who recently strutted down the Victoria’s Secret Fashion Show runway.

“He has been spotted dropping her off at the airport with kisses as she is about to take off for another job,” says the source. “They are just a very sweet couple.”

According to the source, the couple also often hosts Cooper’s mother, Gloria, at the three-bedroom, three bath home. “They also are very close with their families. After Bradley’s dad passed away (in 2011,) his mom Gloria mostly lives with Bradley in L.A.”

The extra roommate doesn’t seem to be a bother to Shayk. “Irina was quickly introduced to Gloria and the two seem to get along great,” the source tells PEOPLE. “They even spend time without Bradley. They take walks, go shopping, or share long lunches.”

If the couple stay in their current home once baby arrives, it’s going to be a full house!

For more on Bradley Cooper and Irina Shayk, pick up the latest issue of PEOPLE, on newsstands Friday.