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7 year oldThe pair appeared in front of thousands of fans at a news conference in Los Angeles - the first of four events called to promote the Las Vegas fight.
"The movement, power, ferociousness - he has not experienced this," said UFC lightweight champion McGregor, 28.
Mayweather replied: "I guarantee you're going out on your face or your back."
The former five-weight world champion, 40, said he was an "old man" but "still had enough" to beat McGregor, who has never boxed professionally.
He teased McGregor, claiming he "only made $3m in his last fight" and referenced the Irishman's submission defeat by Nate Diaz in 2016, his only loss since joining UFC three years earlier.
"We know 'Mr Tap Out' likes to quit and you will wave that white flag," added Mayweather.
McGregor's suit for the event contained pin-stripes which were comprised of an offensive two-word insult.
He was first to break the silence as the pair stared at one another following lengthy walks to the stage. He began dancing in front of Mayweather before both addressed the crowd with expletive-laden speeches.
"His little legs, little core, little head, I'm going to knock him out inside four rounds, mark my words," said McGregor. "I'm a young, confident, happy man that has worked extremely hard for this."
Dublin-born McGregor said the boxing match offered "a limited set of rules" and pointed to demands he says Mayweather's team have made in agreeing to a fight which will reportedly earn each man around $100m (£78.5m)
He added: "All these rules and restrictions - the gloves have to be 10oz, give him the gloves, no problem. No Mexican-made gloves, no problem. No gloves made of horse hair, fine. All I need is a gum shield, line me up to the ring and I'm good to go."
The fight has drawn criticism from some within boxing given McGregor's lack of record within the sport.
As both men took to the stage, their appearance was the top trend globally on Twitter.
Mayweather retired from boxing after taking his record to 49 fights unbeaten with victory over Andre Berto in 2015.
He has held 11 major world titles but has not stopped an opponent inside the distance since 2011, a point McGregor referenced on stage.
The two men will appear at further promotional events in Toronto, Canada, on Wednesday, Brooklyn, New York, on Thursday and London on Friday.
<p>The two leaders have discussed the Ukraine conflict, with the German chancellor calling on Moscow to hold peace talks with Kiev</p>