"There was something about putting this hat on that made me feel like Superman", West said, as part of a lengthy monologue that touched on topics such as his mental health to employment, prison reform, and more.
Asked if West could be a future presidential candidate, Trump said, "Could very well be".
He went on, "People expect that if you're black you have to be Democrat".
He said his "Make America Great Again" hat gives him power.
Much like his idol, Kanye kvetched about Saturday Night Live, insisting that satirizing the president puts America's status in the world at risk, rather than the president a guy who gets laughs in his face at the United Nations. He praised Trump for being the "freshest", the "flyest" president of the US.
Trump said West is "a smart cookie".
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Mr. West said his top priority was to dissuade Mr. Trump from promoting "stop-and-frisk" police tactics in Chicago. So the fathers lose the jobs and they say we'll give you more money for having more kids in your home.
However, in front of reporters, West went off on a litany of subjects, including how his "Make America Great Again" hat makes him feel like Superman. "We have the right to bear arms", West said.
"That was quite something", Trump said at the end Kanye's freewheeling soliloquy. "That's from the heart". In a weird performance last month on "Saturday Night Live" he delivered an unscripted pro-Trump message after the credits rolled.
Trump welcomed West to the White House on Thursday afternoon, saying the rapper "has been a friend for a long time".
Trump is notoriously unpopular with black Americans and has been accused of being racist.
"This is what our president should be flying in", West said, adding that he wants Apple to help him develop the aforementioned plane. "And you made a Superman cape for me".
The Republican president has advocated prison reform and said he was open to changes in sentencing guidelines, an issue that divides the Republican Party between "law and order" hardliners and moderates.
The rapper's wife, reality star Kim Kardashian, visited the White House in May. West chipped in to say "Only after, it would have to be 2024". Also attending is former National Football League great Jim Brown, a civil rights leader who has met with Trump previously, and Jared Kushner.