If true, their baby name of choice is beyond our wildest expectations.
Her full name is apparently Kaidence Donda West.
According to the Parentdish baby name finder. Kaidence means rhythm, which is sort of apt.
A nurse at the Cedars Sinai hospital where Kim gave birth to their first child on Saturday June 15 let slip the eagerly awaited name. Apparently the baby's middle name is a tribute to Kanye's late mother, Donda.
Earlier today, Twitter was chirping with news the couple had called their daughter - who was born five weeks early - Kai Georgia Donda West. Georgia was thought to represent where Kanye grew up as well as Kim's late dad, Rob Kardashian's middle name George.
We have to admit we're a bit disappointed - we liked the idea of WAG Coleen Rooney's son Kai and reality TV show Kim's daughter sharing the same name.
Despite the reports, there is still no official statement from Kimye regarding their baby's birth or name. Sister Khloe Kardashian tweeted: "I can not even begin 2describe the miracle that is now apart of our family. Mommy/baby are healthy & resting. We appreciate all of the love.
"More info will come when the time is right! Thank you all for understanding! We love you all dearly! Overwhelmed with love right now ♥"
It was thought that the name would be kept secret until formidable Kardashian matriach Kris Jenner's new chat show debuted on July 15. But that's a long time to wait.
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