The 32-year-old American retained his WBC heavyweight title in remarkable fashion with a first-round knockout of Bermane Stiverne on Saturday and quickly set his sights on a unification bout with Joshua, who currently holds the IBF, WBA and IBO titles.
Joshua has previously suggested that he has other options available too, while a fight between Wilder and Dillian Whyte has also been mooted, but Wilder believes that anything else would be a "distraction" to the "most massive heavyweight title fight in history".
"In America we have a lot of up and coming guys – we have a lot of current guys right now – so I don't want people to think that Joshua was my only option nor am I Joshua's only option. Like I said we are kings of the division, we have options. We need each other but we don't need each other," he is quoted as saying by talkSPORT.
"We need each other to make the most massive heavyweight title fight in the history of time. But if he don't want to fight then we can still go on with our lives and we can still do what we do. But I think people are going to demand that fight, the fans want that fight and Joshua has already given his word saying, if the fans want it, he wants it. Sometimes we only have our word and I am a man of my word.
"I have always said what I wanted to do and how I was going to do it – and I done it. Now it's time for him to be a man of his word. When you look at Dillian Whyte, why do we have to go through a lower standard of fighter? Even when you look at some of the opposition that I have fought in the past what's the difference between him and those opposition?
"Dillian Whyte is just a distraction between me and Joshua and everybody knows it. For that reason I am not interested in it. The only way we would be interested in it is if Joshua comes with it."
Wilder has won all 39 of his professional bouts, with 38 of those coming via knockout.