TMZ first reported the purchase.
West's $4.5m new home is reportedly just across the street from Kardashian's in Los Angeles. The couple have four children together.
The five-bedroom home is just over 3,650 square feet, and reportedly cost West $421,000 over the home's asking price. It includes a swimming pool and a stable for horses.
The couple originally lived in Kardashian's house together, which West designed over a number of years. After the couple split, West gave his interest in the home to Kardashian.
This is one of several real estate transactions West has made in recent months. In September, the musician purchased a $53.7m beach house in Malibu. He has also recently listed another one of his properties for $3.7m.
The purchase may be part of West's continued efforts to repair his relationship with Kardashian, who is reportedly currently dating SNL comedian Pete Davidson.
During a recent concert with Drake at the La Memorial Coliseum, he changed the lyrics to a song, singing that "I need you to run right back to me, more specifically Kimberly."
In November, West went on a rant claiming that God wanted him and Kardashian to be together.
“The narrative God wants is to see that we can be redeemed in all these relationships,” he said. “We’ve made mistakes. I’ve made mistakes. I’ve publicly done things that were not acceptable as a husband, but right now today, for whatever reason — I didn’t know I was going to be in front of this mic — but I’m here to change the narrative.”
At the time, West said he has to be "next to my children as much as possible," which may explain the purchase of the home near Kardashian's.
West's attempts to rekindle his relationship with Ms Kardashian have not been entertained by the reality TV star.
Kardashian has filed for divorce, and stated in her court documents that "no counselling or reconciliation effort" can fix the couple's marriage. She added that "irreconcilable differences" continue to exist between the pair.
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