The star, who has legally changed his name to Ye, posted a video of his campaign logo to social media, alongside the caption Ye 24.
He also claimed to have asked Donald Trump to be his running mate.
West previously ran for president in 2020, but that campaign flopped, attracting a meagre 70,000 votes.
His latest claims came in a video posted after West was spotted at Trump’s Mar-A-Lago golf club earlier this week, accompanied by Nick Fuentes, a prominent white nationalist.
West said his request for a running mate left the former president, who recently launched his own re-election campaign, “most perturbed”.
In a video titled Mar-A-Lago Debrief, West claimed: “Trump started basically screaming at me at the table, telling me I’m going to lose. Has that ever worked for anyone in history?”
The launch of West’s campaign comes as the rapper faces a number of damaging controversies.
He provoked a storm of criticism after attending Paris Fashion Week in a T-shirt bearing the slogan “White Lives Matter” – a phrase adopted by white supremacists, who began using it in 2015 as a response to the Black Lives Matter movement.
West then claimed his critics were being paid by a secret cabal of Jewish people, a common trope of antisemitism.
As he continued to make antisemitic comments online and in television interviews, the 45-year-old was dropped by his talent agency, while fashion companies including Gap, Adidas and Balenciaga said they would no longer work with him.
The musician later commented that he had lost “two billion dollars in one day”.
Earlier this week, Rolling Stone magazine reported claims that West had used “porn, bullying and mind games” to create a “toxic environment” among Adidas employees working on his Yeezy brand shoes.
The company said on Thursday it had launched an independent investigation into the claims.
When West ran for president in 2020, he announced his campaign too late to appear on the ballot in at least six states.
He held only one rally, in which he broke down in tears as he discussed abortion, and funded two television adverts. In the end, he was only listed as a candidate in 12 states.
For his 2024 bid, the rapper suggested he had enlisted alt-right commentator Milo Yiannopoulos as his campaign manager.
A former editor at the right wing publication Breitbart, Mr Yiannopoulos was largely shunned by mainstream conservatives after a video emerged in 2017 of him appearing to condone paedophilia. He said the comments were “gallows humour” and stated his “disgust” at the sexual abuse of minors.
Most recently, Mr Yiannopolous worked as an intern for Republican congresswoman Marjorie Taylor Greene.