West, who lavished praise on the president ahead of a planned lunch, launched into a near 10-minute monologue that Mr Trump called “quite something”.
Referring the Make America Great Again hat he was wearing while sat across from Mr Trump in the Oval Office he said that it felt like it gave him “power”.
“There was something about putting this hat on that made me feel like Superman,” West said, as part of the speech that spanned topics from his mental health - saying he did not have dipolar disorder, but was merely “sleep deprived” - and violence in Chicago.
The rapper has been criticised over his for his support for Mr Trump, including after a recent appearance on NBC's Saturday Night Live.
“People expect that if you're black you have to be Democrat,” he said. “What I need Saturday Night Live to improve on, what I need the liberals to improve on: if he don't look good, we don't look good,” West said, gesturing at Mr Trump.
Taking out his phone, he showed the president an image of a hydrogen-powered plane and suggested the president should use the design for Air Force One.
West also said he would like Mr Trump to ditch talk about persuading police in Chicago to institute “stop and frisk” policies to crack down on crime and violence.
“I didn't mean to put you on the (spot), bro,” West told Mr Trump, who said “I'm open to everything” about the subject, adding that West could speak for him “any time” as he was a “smart cookie”.
West also said the president is on a “hero's journey.”
“Let's stop worrying about the future, all we have is today,” the music star said, commenting on his own presidential ambitions. “Trump is on his hero's journey right now. He might not have thought he'd have a crazy mother-f***** like me [supporting him].”
Asked if he saw West as a potential presidential candidate Mr Trump said “could very well be” with West adding “only after 2024” - the end of a potential second White House term for Mr Trump.
Over lunch, according to White House spokesman Hogan Gidley, Mr Trump and his guests - which include former NFL running back Jim Brown and White House advisors Jared Kushner and Ivanka Trump - would discuss potential future pardons for convicts along with “addressing the massive violent crime surge in Chicago” and efforts to create jobs for black Americans.
Kim Kardashian-West, who is married to the rapper, had previously raised the issue of pardons in a White House meeting of her own.
Before leaving the Oval Office for the lunch of roast chicken and potatoes West stood up and gave Mr Trump a hug, saying: “I love this man”.
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