Joshua holds the IBF, WBO and WBA belts and any bout between the two would unify the heavyweight division, BBC Sport reported.
Earlier this week Wilder, who drew with Britain's Tyson Fury in Los Angeles earlier this month, accused Joshua of "begging" for a fight.
"I'm not interested in Tyson Fury – he's not the champion," Joshua said.
The 29-year-old is yet to confirm an opponent for his next fight on April 13 at Wembley Stadium, with his promoter Eddie Hearn saying "it comes down to what Wilder wants to do".
Joshua told ESPN, "I'm willing to fight Deontay Wilder on April 13 in London. I don't know what more I need to do to get that message across. He's more interested in fighting Tyson Fury. Fury holds no world titles.
"I've booked the date in advance, booked the venue in advance. I want to make my point clear in front of everyone – I'm willing to fight any one of these guys, especially the champion Deontay Wilder.
"I'm the one leading the pack since I made my debut. Five years in, several title defenses and I'm running the game. I mean business. I don't know what more I need to do to get that message across to the world. I can handle my business and I'm willing to do that in the ring."
Following the draw between Wilder and Fury earlier this month the WBC sanctioned a direct rematch.
Joshua's promoter Hearn has previously engaged in failed negotiations with Wilder's team.