The couple have been married for almost seven years and have four children together.
The news was broken by celebrity news website TMZ after months of rumours of marital difficulties.
Reports in US media suggest the reality star, 40, has requested joint legal and physical custody of their children. Neither has publicly commented.
The couple are among the most recognisable stars in the world and are both hugely successful in their own right.
Kim first found fame in 2007 as the star of an E! Television reality series about her family. Keeping up with the Kardashians has remained hugely popular since, with its 21st and final series due to air next year.
The reality star has found success in many other areas of business, from mobile apps to make-up, and Forbes estimates her personal wealth to be about $780m (£556m).
Known for huge international hits like Gold Digger, Kanye West has been one of the biggest names in rap music for over 15 years. The Grammy-winning musician has also found success as a fashion designer.
The pair were friends for years before they got together and West first appeared on her family’s reality show in 2010.
Their first daughter, North, arrived in June 2013.
West famously hired the San Francisco Giants’ baseball stadium and an orchestra to propose to her in front of her family, and their show’s cameras, later that year.
They got married in Italy in May 2014 and a photograph of them kissing on their wedding day became, at the time, Instagram’s most liked post ever. By this point she and her sisters had turned social media into an art form, with their online followings propelling their business success and fame ever further. (Courtesy BBC)