He's 'Excited but Nervous!' How George Clooney Is Getting Ready to Be a Dad to Twins

When Amal Clooney visited her husband on the set of his upcoming film, Suburbicon, late last year in L.A., the Oscar winner displayed an unmistakable sense of pride.

“George and Amal were both glowing,” a set source tells PEOPLE in the new issue. “There were hints and rumors about her being pregnant. She seemed careful with food and didn’t have any caffeine.”

Always gracious, Amal “greeted everyone with a friendly handshake and a huge smile. She acted very down-to-earth,” the source adds. “She didn’t ask for anything and just wanted to hang out in the background. George wouldn’t really let her, though. He would bring her around the set, beam and introduce her as ‘my wife, Amal.’ It was very cute to see how excited he was.”

Now the world knows that the former perennial bachelor, 55, who stunned everyone in 2014 by proposing to Amal, a Lebanese-British human-rights lawyer, is about to become a first-time father — times two.

Multiple sources confirmed to PEOPLE that Amal, 39, is expecting twins — a boy and a girl — this spring.

“George is excited but nervous,” says a source close to the couple’s inner circle. “He is excited about the kids, but also has the normal amount of nerves that come with being parents.”

For her part, Amal “strikes me as someone who’ll be a fantastic mother,” says another insider. “Intelligent, loving, funny, smart, nurturing. She doesn’t seem someone who will let kids get away with being Hollywood children.”

With houses in L.A. and Lake Como, the couple has been spending more time at their newly-renovated mansion just outside of London that they purchased for a reported $12 million shortly after their wedding in September 2014.

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Both have continued to work but Amal “stopped traveling for dangerous missions months ago,” says another source. Clooney “knows how brilliant Amal is in her work and how nurturing of a person she is,” adds the source close to the couple’s inner circle. “He knows she will be a wonderful mother.”

As for the actor, he “hinted that he might be taking some time off,” says the Suburbicon source. “That would be very rare for him, but now it all adds up. He’s going to be the best dad.”

For much more about how the Clooneys are getting ready for parenthood, pick up this week’s issue of PEOPLE, on newsstands Friday.