Joshua won back the IBF, WBA and WBO titles from Andy Ruiz Jr in Saudi Arabia last week, avenging his shock defeat at Madison Square Garden six months earlier.
The only strap not in Joshua's collection is the WBC title, which American Wilder has held since 2015.
Wilder is due to face Fury in a rematch on February 22, and Joshua wants to fight the winner in an undisputed bout.
"I feel like it's 'The Lord of the Rings', the last ring, let's get it on, let's make history," he told BBC Radio 2.
"It has to happen in 2020, we're in the same division, same era, if these guys want to make history - especially Wilder, because he has the last belt.
"We created history by becoming a two-time champion of the world, and I feel like we can add to that legacy in 2020."
Joshua later wrote on Twitter: "Let's get 'undisputed' trending! That's what I want & that's what the people want... I want it bad! I came to take over & not take part!"
Ruiz has spoken of his desire for a third fight with Joshua, though his poor performance in the Briton's points win means that is unlikely.
Instead, Joshua has his sights set on the WBO's mandatory challenger Oleksandr Usyk - who stepped up to heavyweight earlier this year after becoming an undisputed champion at cruiserweight - or his IBF equivalent Kubrat Pulev.
"If I don't fight those guys, I'll have to give up my belt, and I'd rather fight to defend them," Joshua added.