The Bronze Bomber made light work of opponent Dominic Breazeale in New York on Saturday, knocking him out in the first round to retain his WBC Heavyweight title.
Attentions have now turned back to fixing a fight with Joshua – who will first face Andy Ruiz Jnr on June 1.
And Wilder was adamant this was one bout which simply must be made.
“I don’t want this to be a Lennox and Bowe situation,” Wilder told Sky Sports.
“The heavyweight division is on fire at the moment and it is our obligation to make the fight happen for the fans.
“I think the next time the negotiations come around something will get done.
“I promise you that we will make it happen. It won’t be a Lennox Lewis and Riddick Bowe situation.”
Lewis and Bowe fought in the final of the 1988 Olympic Games, with the former coming out on top.
However both men avoided each other in their professional careers in what is deemed as one of the greatest fights that never happened.
The same risk now presents itself with Joshua and Wilder, with a unification fight between the pair the most eagerly anticipated fight in recent heavyweight boxing history.
Both fighters remain unbeaten, with Wilder having won 40 of his 42 bouts by knockout, compared with Joshua’s 21 in 22.
Wilder, who drew with Tyson Fury last year, said before the fight with fellow American Breazeale that he still wanted a “body on his record”.
He was widely criticised for the comment.
But he added: “I don’t regret nothing I said but I’m glad what I said didn’t happen and he can go home with his family.”