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Celebrity magazines Us Weekly and People quoted sources close to the couple as saying the surrogate was pregnant and expected to give birth in May.
Representatives of the reality TV star and her rapper husband did not respond to a request for comment.
Kardashian and West's third child, Chicago, was born via a surrogate in January 2018. The couple has two other children North (5) and three-year-old Saint.
They turned to surrogacy after doctors warned Kardashian of serious health risks if she became pregnant again after her first two children.
Us Weekly reported that the couple put their plans for a fourth child on hold about six months ago as West was struggling with mental health issues.
The rapper said last year he suffers from bipolar disorder and embarked on a bizarre Twitter spree that culminated in a rambling October meeting at the White House with US President Donald Trump, whom West supports.
"Kim wanted their carrier to be pregnant shortly after Chicago’s birth, but they put those plans on hold when Kanye was having his issues,” an anonymous source was quoted as telling Us Weekly.
"Kanye has been great. He’s been healthy, happy and his head is clear. They are both very excited for this new baby," the source added.
West, who said after the Trump meeting that he was distancing himself from politics, kicked off the New Year with a series of tweets reiterating his support for the Republican president.
"Trump all day," the musician wrote on Twitter on January 1, saying he planned to perform wearing the "Make America Great Again" hat made popular by Trump and his supporters.
"One of my favorite of many things about what the Trump hat represents to me is that people can’t tell me what to do because I’m black," West added.