A source told People magazine that the rapper, 43, "definitely understands that he upset Kim. He feels very bad about it. It's obvious that he still loves Kim."
The insider added that West "seems much more relaxed and chilled out for the past couple of days" and has "regrets about sharing such private details" about his reality TV star wife, 39.
West took to Twitter on Saturday and issued an apology to Kardashian, days after he revealed he had considered divorcing her.
"I would like to apologize to my wife Kim for going public with something that was a private matter. I did not cover her like she has covered me.to [sic] Kim I want to say I know I hurt you. Please forgive me. Thank you for always being there for me," the Yeezy designer wrote.
Although West did not specify what "private matter" he was referring to, it's likely related to his admission during a South Carolina campaign rally last weekend that the couple considered abortion when they were expecting their first child, daughter North, 7.
"Even if my wife wants to divorce me after this speech, she brought North into the world even when I didn’t want to," West told a crowd while crying. "She stood up and she protected that child. You know who else protected a child? Forty-three years ago, who do you think protected a child?”
Reports indicated earlier in the week that Kardashian was "furious" over the abortion claims.
West then claimed in a now-deleted tweet that he's been "trying" to divorce Kardashian ever since she attended a prison reform event also attended by rapper Meek Mill.
The Yeezy founder also subtly shaded the reality star's past photo shoot with Playboy magazine and claimed his mother-in-law, Kris Jenner, was no longer allowed to see the couple's four kids.
Kardashian ultimately broke her silence on the drama on Wednesday with a public statement acknowledging West's bipolar disorder.
"As many of you know, Kanye has bi-polar disorder. Anyone who has this or has a loved one in their life who does, knows how incredibly complicated and painful it is to understand," Kardashian's statement shared to Instagram began.
Kardashian said she realized that her husband’s words can carry particular weight and “cause strong opinions and emotions” due to his status as a public figure. However, she said she finds him to be a “brilliant and complicated” person that is simply struggling to deal with his mental illness at times.
Fox News' Melissa Roberto contributed to this report.