Kim Kardashian and Kanye West 'Have a Name in Mind' as Newborn Son Meets Siblings: Source

Fans of Kim Kardashian West and Kanye West might not have to wait long to find out the name they’ve chosen for their newborn son.

“They have had a name in mind, but wanted to meet the baby before they decided,” a Keeping Up with the Kardashians source tells PEOPLE of the latest addition to the family, whose arrival via surrogate Kardashian West, 38, announced on Twitter Friday.

“Kim and Kanye were both present for the hospital birth on Thursday,” the insider adds. “They are ecstatic. Their little boy is super cute and looks like Kim.”

According to the source, the couple’s three older children — daughters Chicago, 15 months, and North, 6 next month, plus son Saint, 3 — “have met their brother,” and “North seems the most excited.”

“The baby is still at the hospital today, but it seems he will be home before Mother’s Day,” the insider says.

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Is This the Sweet Family Name Kim Kardashian and Kanye West Will Give Their Son? Here Are the Big Clues

The reality-television veteran confirmed on Twitter on Friday that she and her rapper husband, 41, had welcomed their fourth child, a son, via surrogate.

“He’s here and he’s perfect!” Kardashian West tweeted, following up minutes later with a second post that read, “He’s also Chicago’s twin lol I’m sure he will change a lot but now he looks just like her.”

For now, this physical descriptor will have to satisfy fans’ curiosity as the couple have yet to share any photos of their newborn baby boy, whose imminent arrival PEOPLE confirmed on Jan. 2.

20 Name Suggestions Kanye and Kim Kardashian West Should Definitely Consider for Their Newborn Son

News that the couple’s surrogate was in labor came Thursday on The Ellen DeGeneres Show, when Kourtney Kardashian stopped by during mom Kris Jenner‘s appearance for a surprise visit.

Along with quite the brood — her three children (Reign Aston, 4, Penelope Scotland, 6½, and Mason Dash, 9) plus Chicago, Saint and North — Kourtney, 40, brought the news that Kardashian West’s surrogate would be giving birth shortly, explaining why her sister wasn’t on the show as planned.

“What! What are we doing here?!” said Jenner, 63, visibly shocked, adding, “For how long?”

“And nobody knows, so everyone in the audience don’t tell,” Kourtney explained while laughing.

“He’s Here!” Kim Kardashian and Kanye West Welcome Son: “He’s Perfect”

The couple have yet to reveal their baby boy’s name, but in April, the KUWTK star shared that while she’d initially wanted an Armenian name for the couple’s second son, she hadn’t found anything she felt great about yet.

However, she did like the idea of naming her new baby after Rob Kardashian. “I was truthfully thinking about just naming him Rob, after my brother, Rob,” she shared during an episode of Jimmy Kimmel Live! in Las Vegas, revealing that Rob, 32, had already given her his blessing.

In addition to her brother, the name was also shared by Kardashian West’s late father, Robert Kardashian.

“My brother approved it, so that’s like our one kind of name,” she continued, before admitting she still had her reservations about the choice as it was very different than her other children’s names.

 20 Name Suggestions Kanye and Kim Kardashian West Should Definitely Consider for Their Baby Boy on the Way

The name “Rob” popped up again weeks later, though, when Kardashian West celebrated her baby shower.

During the KKW Beauty mogul’s “CBD & Meditation”-themed bash, guests got a chance to suggest unique monikers for baby West — and the star chose one name in particular to highlight.

Although there were clearly many options on the board, Kardashian West zeroed in on a note on the wall that was suggesting “Robert Harut.”

The note went on to reveal that “Harut” is an Armenian word, meaning “he has risen” — and as Kardashian West had previously indicated that she was into the idea of naming her son Rob and finding an Armenian name, the choice seemed like a perfect fit.

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