Joshua was present on American soil to promote the defence of his IBF, WBA and WBO titles against Russian Alexander Povetkin on September 22 at Wembley Stadium.
However, Miller interrupted the head-to-head meeting between the two heavyweights to confront the unified champion, who responded to the goading by turning to face up to the trash-talking American.
The pair were quickly separated on stage by promoter Eddie Hearn but, asked in a video interview with Sporting News if he saw fear in Joshua's eyes during their head-to-head, Miller replied: "Most definitely. Most definitely.
"Nobody brings it to him like that. He tried to shut down [Dominic] Breazeale and some other guys who talked s***, but I ain't the one, I ain't the one.
"I would have flipped him on his head if there was nobody in between us."
Miller was present as one of a number of new signings by Matchroom following their rights partnership with DAZN, with IBF light-heavyweight holder Artur Beterbiev, WBA super-bantamweight title-holder Daniel Roman and two-weight world champion Jessie Vargas also among the list of recruits.
While he will have to wait for his shot at Joshua, the fighter nicknamed 'Big Baby' is ready and willing to travel for a crack at the undefeated Briton.
— Matchroom Boxing (@MatchroomBoxing) July 17, 2018
"I will go anywhere to fight him next – I'm ready to bust him up," he added.
Joshua responded to the incident by warning Miller - who boasts a 21-0-1 record as a pro - not to bite off more than he can chew.
"With Miller, he's always been very negative towards me," Joshua told Sporting News.
"I'm not in the business of talking just for clout. I talk because it's real. If I believe I’m better than you, I'm saying that because I am. I'm not just talking it because I want to be popular.
"He started talking behind my back. It's a territorial thing – if you're going to bark, I'm going to roar."
Photo Getty Images. Caption: American heavyweight Jarrell Miller