Britain's Joshua reclaimed his IBF, WBA and WBO titles from Andy Ruiz Jr. in Saudi Arabia last Saturday, BBC Sport reported.
WBC champion Wilder is scheduled for a rematch with Britain's Fury on February 22.
"I feel like it's 'Lord of the Rings', the last ring, let's get it on, let's make history," said Joshua, 30.
Speaking on ta BBC Radio 2 show, he added, "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."
By defeating Ruiz, Joshua joined a small cluster of men including Muhammad Ali, Lennox Lewis, Evander Holyfield, Mike Tyson and Floyd Patterson to have reclaimed the world heavyweight title.
Joshua said he would fight either the WBO's mandatory challenger – currently Ukraine's Oleksandr Usyk – or the man in the same position with the IBF, Bulgarian Kubrat Pulev next.
"If I don't fight those guys I'll have to give up my belt and I'd rather fight to defend them," he added.