The move came as it emerged the star had shared a series of comments based on racist conspiracy theories in a separate interview with Fox News.
Fox removed those segments before broadcast, but the footage was leaked to technology website Motherboard.
West has not commented on either case.
The BBC has asked both him and Fox News for a response.
The star, who is legally known as Ye, was previously suspended from Instagram and Twitter for making anti-Semitic comments.
Those posts came in response to a backlash against his show at Paris Fashion Week, where he wore a t-shirt carrying the slogan “White Lives Matter”.
The Anti-Defamation League has called the phrase “hate speech” and attributed it to white supremacists, who began using it in 2015 in response to the Black Lives Matter movement.
Amid the backlash, he appeared on Fox News, where he told host Tucker Carlson the t-shirt was “funny” and “the obvious thing to do”.
In unaired clips from the same interview, which leaked on Tuesday, West detailed his belief in an unfounded anti-Semitic conspiracy theory that Planned Parenthood was founded “to control the Jew population” in conjunction with the Ku Klux Klan.
“When I say Jew, I mean the 12 lost tribes of Judah… who the people know as the race Black really are,” he told host Tucker Carlson, referring to a claim, unsupported by historical evidence, that Black people are the “real” Jewish race, and that Jews are attempting to “steal” their birthright.
West also complained that his children were attending a school where Kwanzaa – an annual celebration of African-American culture – is taught, saying he would rather they learned about the Jewish holiday Hanukkah because “at least it will come with some financial engineering”, yet another anti-Semitic trope.
In another clip, the rapper confirmed he had received the Covid-19 vaccine, despite previously claiming the shots were “the mark of the beast” and part of a plot to implant chips in people.
He also claimed that “fake children” had been placed in his home to manipulate and “sexualize” his four children with former wife Kim Kardashian.
Fox has not explained why it excluded these clips from its broadcast, although most television interviews are edited and condensed for clarity.
Producers of The Shop: Uninterrupted on the other hand, decided against airing their interview with West altogether, and said they would not share details of his comments.
Maverick Carter, who produces the show with basketball legend LeBron James, told the Andscape website: “Kanye was booked weeks ago and, after talking to Kanye directly the day before we taped, I believed he was capable of a respectful discussion and he was ready to address all his recent comments.
“Unfortunately, he used The Shop to reiterate more hate speech and extremely dangerous stereotypes.”
The statement continued: “While The Shop embraces thoughtful discourse and differing opinions, we have zero tolerance for hate speech of any kind and will never allow our channels to be used to promote hate.
“I take full responsibility for believing Kanye wanted a different conversation and apologise to our guests and crew. Hate speech should never have an audience.”
It is believed that James was not present as the interview took place.
West was diagnosed with bipolar disorder years ago and has publicly spoken about his challenges with his mental health.
However, medical experts and people who share West’s condition have warned that mental health problems do not go hand-in-hand with anti-Semitism.
“There are many people who don’t have mental health issues who are racist and bigoted. And there are people with mental health issues who are not racist or bigoted,” clinical psychologist Carla Manly told USA Today. “We want to see those as two very different issues.”
“I think Kanye is honestly just an idiot,” Sam, who has bipolar disorder, told BBC World Service.
“He rolls with his disorder and lets it harm whoever is around him, and I think that’s extremely irresponsible.
“I don’t think he should be associated with bipolar disorder or the mental health movement, because he doesn’t speak for us.”