Wilder came face-to-face with Joshua as he walked past the Briton after the fight. The two fist-bumped, and the American said, "Let's do it."
"It wasn't a few words, I dapped it up with him and I said, now let's get it on," Wilder told reporters when asked about the exchange.
"I'm ready for whoever, long as these guys are in the heavyweight division, I am here. You can call me Mr Clean, because I want to clean up the whole division. The division is nothing without Deontay Wilder."
American Wilder was the WBC champion when Joshua held the WBA, IBF and WBO belts, but a unified heavyweight championship bout between them never materialised, as Wilder went on to lose to Tyson Fury and Joshua lost his belts to Oleksandr Usyk.
Undefeated Usyk went on to become the unified champion when he beat Fury in May 2024.
Joshua was last in action in December, when he knocked out Jake Paul.
A few days later, he was hospitalised following a car accident in Nigeria that killed two of his close friends.
Joshua's promoter Eddie Hearn, who was with him during the exchange with Wilder, said the fight would not be a problem for the Briton.
"He said, let's do it. AJ sort of stared at him ice cold, but he would fight him no problem," Hearn told Fight Hub TV.