The undefeated American, whose record stands at 37 knockouts from 38 fights, is desperate to get into the ring with the IBF, WBA and IBO heavyweight champion.
Joshua is currently weighing up his options with potential fights against Joseph Parker and Tyson Fury also in the pipeline.
But Wilder has taunted the Londoner by saying he has “had it easy” in making it to the top.
“Joshua, let’s make this fight happen man,” Wilder told ESPN.
“This is one of the biggest, most exciting fight in the world history.
“We could make records, we could set history on the time of the fight whenever it happens.
“I’m ready, I don’t know about you. You’ve had it easy, it’s been breezy for you.
“You’re ready to fight low opposition and get paid millions of dollars and you don’t want that gravy train to end, but I am the man that’s going to end your career.”
Joshua’s last fight was against Carlos Takam on October 28, while Wilder stopped Bermane Stiverne in the first round a week later.
And talks have began on uniting the two great heavyweights, but it is yet to be seen if they reach can agreement.