Others were not convinced, offering instead 'Mexico,' 'Don,' 'Cisco,' and even 'Wild' which would make the boy 'Wild West.'
An insider told People on Friday that the name has been chosen.
'They have had a name in mind, but wanted to meet the baby before they decided,' a source said.
'Kim and Kanye were both present for the hospital birth on Thursday. They are ecstatic. Their little boy is super cute and looks like Kim. The baby is still at the hospital today, but it seems he will be home before Mother's Day.'
The 38-year-old pinup has yet to release the name of the little one. The boy already has North, aged five, Saint, aged three, and Chicago, aged one, as siblings.
But her 60M Twitter followers think they have it all figured out after the Selfish author shared a message with a special emoji over the weekend.
The KKW Beauty mogul posted an image of a teddy bear during her recent note.
Fans think that could mean the name of the child is Bear. Another possibility is Teddy.
Bear West and Teddy West both have a ring to them, but her followers could be making something out of nothing as it was only an emoji.
The note read: 'We celebrated our baby boy about a week ago and now he’s here!
'He’s so perfect! Here are some pics of my CBD baby shower,' began the Vogue cover girl.
'I was freaking out for nothing because he is the most calm and chill of all of my babies so far and everyone loves him so much.'
Then the KUWTK star added the image of the cute bear that had a red bow around its neck.
Fans said, 'I think she named her son Bear' and another added, 'Oh yeah it's Bear all right, she just dropped dat.'
And the kids have met baby four as well. According to the source, Chicago, aged one, Saint, aged three, and and North, aged five, 'have met their brother' and 'North seems the most excited.'
In April, the fashion mogul told Jimmy Kimmel that she was looking for an Armenian name for the boy, but that she had not found a name she liked yet.
Typical Armenian names for a boy are Aram, Garik, Avet, Sergey, Grigor, Vahan and Alen, to name a few.
She then revealed that she was thinking of keeping the name in the family.
'Actually, I do have one name. I was truthfully thinking about just naming him Rob, [after] my brother,' Kim said on Jimmy Kimmel Live!
She also said that Rob, 32, had given her his blessing to use his name.
'My brother approved it, so that's like our one kind of name,' the Keeping Up With the Kardashians star said.
Kim's late father was also a Rob, or Robert, to be more specific. He died in 2003 from cancer.
But she's not confident about the name, as it's pretty different from her other kids' names.
The entrepreneur also tweeted that her newborn looked like 'Chicago's twin lol I'm sure he will change a lot but now he looks just like her.'
No word on when the media personality will share pics of the newborn, but if the timeline is anything similar to when she's shared photos of her babies in the past, it will likely be at least a month.
Also this weekend, Kim shared images of roses that had writing on the petals. One said, 'It's a boy.' And another said North, for her first child North West.
There was one also that said Saint, her first boy. And the last one was for Chicago, who was her first child born via gestational carrier.
© Associated Newspapers Ltd.